The document discusses the concept of slope of a line. It defines slope as the ratio of the "rise" over the "run" between two points on a line. Specifically:
- The slope of a line is calculated as the change in the y-values (rise) divided by the change in the x-values (run) between two points on the line.
- This formula is easy to memorize and captures the geometric meaning of slope as the tilt of the line.
- An example problem demonstrates calculating the slope of a line between two points by finding the difference in their x- and y-values.
Okay, here are the steps to solve this problem:
* Florante's location is given as point (2, 1)
* Laura's x-coordinate is given as 8
* The slope of the line containing their locations is given as 1/3
* We can use the slope formula to find Laura's y-coordinate:
Slope (m) = (Change in y) / (Change in x)
* Change in x from Florante to Laura is 8 - 2 = 6
* Change in y is what we want to find and is represented by y - 1
* We are given: m = Rise/Run = 1/3
* Plugging into the slope formula:
1/
1) The document discusses the differences between linear equations and inequalities in two variables. Linear equations use the equal sign while inequalities use symbols like <, >, ≤, ≥, ≠.
2) The graph of a linear equation is a single line, while the graph of an inequality shows the shaded region that satisfies the inequality. For < or > the line is broken, and for ≤ or ≥ the line is solid.
3) The document provides steps for graphing a linear inequality in two variables: graph the line by changing the inequality to an equation, use a test point to determine which side to shade, and shade the area where the test point satisfies the inequality.
This document discusses linear functions and how to represent them using equations, graphs, and tables of values. It defines a linear function as one that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Examples are provided to illustrate determining the slope and y-intercept from an equation and representing a linear function using an equation, table of values, or graph. It is explained that a linear function will produce a straight line on a graph and have constant differences in x- and y-coordinates in its table of values.
A linear inequality is similar to a linear equation but uses inequality symbols like < or > instead of =. A solution to a linear inequality is any coordinate pair that makes the inequality true. A linear inequality describes a half-plane region on a coordinate plane where all points in the region satisfy the inequality, with the boundary line given by the related equation. To graph a linear inequality, you solve it for y, graph the boundary line as solid or dotted, and shade the correct half-plane above or below the line.
This document discusses trigonometry and its key concepts. It defines trigonometry as a branch of mathematics concerning the study of triangles and the relationship between side lengths and angles. It notes that Hipparchus of Nicaea in the 2nd century BC is considered the father of trigonometry. The document outlines important trigonometric ratios like sine, cosine and tangent, and defines concepts like adjacent, opposite, and hypotenuse sides of a right triangle. It also discusses Pythagoras' theorem relating side lengths in a right triangle.
This document is a daily lesson log for a Grade 8 mathematics class covering rational algebraic expressions. The objectives are to perform operations on rational algebraic expressions, specifically division. Five examples of dividing rational expressions are provided. The lesson procedures involve reviewing previous concepts, presenting examples, group work and discussion, and a group contest to develop mastery. Formative assessment involves dividing additional rational expressions. The log does not provide any information in the reflection section to evaluate the lesson's effectiveness.
The document defines and compares angle of elevation and angle of depression. When looking at an object above your position, the angle formed between the line of sight and horizontal is the angle of elevation. When looking below your position, the angle is called the angle of depression. Several examples are given of calculating angles of elevation and depression using trigonometric tangent functions, when the height of the observer or object and the distance between them is known.
- The document discusses how computers use negative exponents to process fractions and percentages during photo processing, which allows apps like Photoshop to shrink photos.
- It explains that negative exponents represent fractions, with the item with the negative exponent moving to the denominator when written as a fraction. This allows computers to perform mathematical operations involving fractions.
- Examples are provided of how negative exponents simplify to fractions through the rule of a-m = 1/am, with the item in the exponent moving between the numerator and denominator.
Okay, here are the steps to solve this problem:
* Florante's location is given as point (2, 1)
* Laura's x-coordinate is given as 8
* The slope of the line containing their locations is given as 1/3
* We can use the slope formula to find Laura's y-coordinate:
Slope (m) = (Change in y) / (Change in x)
* Change in x from Florante to Laura is 8 - 2 = 6
* Change in y is what we want to find and is represented by y - 1
* We are given: m = Rise/Run = 1/3
* Plugging into the slope formula:
1/
1) The document discusses the differences between linear equations and inequalities in two variables. Linear equations use the equal sign while inequalities use symbols like <, >, ≤, ≥, ≠.
2) The graph of a linear equation is a single line, while the graph of an inequality shows the shaded region that satisfies the inequality. For < or > the line is broken, and for ≤ or ≥ the line is solid.
3) The document provides steps for graphing a linear inequality in two variables: graph the line by changing the inequality to an equation, use a test point to determine which side to shade, and shade the area where the test point satisfies the inequality.
This document discusses linear functions and how to represent them using equations, graphs, and tables of values. It defines a linear function as one that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Examples are provided to illustrate determining the slope and y-intercept from an equation and representing a linear function using an equation, table of values, or graph. It is explained that a linear function will produce a straight line on a graph and have constant differences in x- and y-coordinates in its table of values.
A linear inequality is similar to a linear equation but uses inequality symbols like < or > instead of =. A solution to a linear inequality is any coordinate pair that makes the inequality true. A linear inequality describes a half-plane region on a coordinate plane where all points in the region satisfy the inequality, with the boundary line given by the related equation. To graph a linear inequality, you solve it for y, graph the boundary line as solid or dotted, and shade the correct half-plane above or below the line.
This document discusses trigonometry and its key concepts. It defines trigonometry as a branch of mathematics concerning the study of triangles and the relationship between side lengths and angles. It notes that Hipparchus of Nicaea in the 2nd century BC is considered the father of trigonometry. The document outlines important trigonometric ratios like sine, cosine and tangent, and defines concepts like adjacent, opposite, and hypotenuse sides of a right triangle. It also discusses Pythagoras' theorem relating side lengths in a right triangle.
This document is a daily lesson log for a Grade 8 mathematics class covering rational algebraic expressions. The objectives are to perform operations on rational algebraic expressions, specifically division. Five examples of dividing rational expressions are provided. The lesson procedures involve reviewing previous concepts, presenting examples, group work and discussion, and a group contest to develop mastery. Formative assessment involves dividing additional rational expressions. The log does not provide any information in the reflection section to evaluate the lesson's effectiveness.
The document defines and compares angle of elevation and angle of depression. When looking at an object above your position, the angle formed between the line of sight and horizontal is the angle of elevation. When looking below your position, the angle is called the angle of depression. Several examples are given of calculating angles of elevation and depression using trigonometric tangent functions, when the height of the observer or object and the distance between them is known.
- The document discusses how computers use negative exponents to process fractions and percentages during photo processing, which allows apps like Photoshop to shrink photos.
- It explains that negative exponents represent fractions, with the item with the negative exponent moving to the denominator when written as a fraction. This allows computers to perform mathematical operations involving fractions.
- Examples are provided of how negative exponents simplify to fractions through the rule of a-m = 1/am, with the item in the exponent moving between the numerator and denominator.
This document discusses integers and absolute value. It defines integers as positive and negative whole numbers and explains how to graph and order integers on a number line. It also defines absolute value as the distance from zero and how to evaluate the absolute value of integers, including that the absolute value of a number can never be negative. Examples are provided for graphing, ordering, and finding opposites and absolute values of integers.
This document discusses how to multiply and divide rational algebraic expressions. It explains that to multiply rational expressions, one multiplies the numerators and denominators separately. Rational expressions must first be factored to cancel common factors before multiplying. Several examples of multiplying rational expressions are shown. The document also explains that to divide rational expressions, one multiplies the numerator by the reciprocal of the denominator. Some examples of dividing rational expressions are provided.
Sample space, events, outcomes, and experimentsChristian Costa
Probability is a branch of mathematics that deals with describing how likely events are to occur. An experiment is a process with uncertain outcomes, while an outcome is any possible result. The sample space is the set of all possible outcomes. Fundamental counting principles state that if one event can occur in m ways, a second in n ways, and a third in p ways, the total number of outcomes is m x n x p. Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
The document discusses factoring the difference of two squares through examples such as (x+5)(x-5)=x^2 - 25. It explains that to factor a difference of two squares, we write the expression as the difference of two terms squared, then group the terms with the same bases and opposite signs inside parentheses. Several practice problems are provided to reinforce this technique for factoring completely the difference of two squares.
An axiomatic system consists of:
1) Undefined terms which are described but not defined;
2) Defined terms which are defined using undefined terms;
3) Axioms or postulates which are considered true without proof;
4) Theorems which are proved using axioms, definitions and logic.
The document discusses various theorems and properties related to triangles, including:
1) The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2) The triangle inequality theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.
3) Properties relating the lengths of sides and measures of angles in a triangle, such as if sides are unequal then angles will be unequal as well.
The document is a lecture on similar triangles. It defines similar triangles as having the same shape but different sizes, and discusses how similar triangles have corresponding angles that are congruent and corresponding sides that are proportional. It provides examples of similar triangles and statements showing their similarity. It also covers using proportions of corresponding sides to solve for missing sides in similar triangles and several proportionality principles related to similar triangles, including the basic proportionality theorem involving parallel lines cutting across a triangle.
This document defines slope and provides examples for teaching students about slope. It explains that slope is the ratio of vertical to horizontal change and can be positive, negative, zero, or undefined. The objectives are for students to identify slope from graphs, calculate slope using rise over run, and apply the slope formula to find slope given two points. Examples are provided to demonstrate calculating slope from graphs and points using rise over run and the slope formula.
This document explains the fundamental counting principle and permutations. It provides examples of how to use the principle to calculate the number of possible outcomes for events with multiple steps or choices. These include examples like counting meal combinations, sandwich varieties, possible license plates, phone numbers, test answers, and card selections. It discusses accounting for situations where items can or cannot be repeated between choices.
This document discusses finding the sum and product of the roots of quadratic equations. It provides the formulas for calculating the sum and product of roots without explicitly solving for the individual roots. The sum of the roots is equal to -b/a, and the product of the roots is equal to c/a, where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. Several examples are worked out applying these formulas. The document also contains exercises asking the reader to find the sum and product of roots for additional quadratic equations using the given coefficients a, b, and c.
Direct variation describes the relationship between two quantities where one quantity varies as the other changes proportionally. It can be represented by the equation y = kx, where k is the constant of variation.
Some key points about direct variation:
- The graph of a direct variation will pass through the origin, as there is no y-intercept term.
- To determine if a relationship represents direct variation, calculate the constant of variation k from the data and check if it remains the same for different values.
- Direct variation can be used to find unknown values by setting up a table with the known values and using the direct variation equation y = kx.
This document provides an introduction to basic probability concepts. It defines key terms like experiments, outcomes, sample space, and events. It explains how to calculate probabilities using fractions, decimals, or percentages. Examples are provided on determining the probability of rolling certain numbers on a die or spinning to certain areas on a spinner. The document also covers concepts like finding the probability of compound events and solving word problems involving probabilities.
12. Angle of Elevation & Depression.pptxBebeannBuar1
This document discusses angles of elevation and depression. It defines an angle of elevation as the angle formed between a horizontal line and the line of sight to an object located above the horizontal line. An angle of depression is defined as the angle formed between a horizontal line and the line of sight to an object located below the horizontal line. The document provides examples of solving problems involving angles of elevation and depression using trigonometric functions like tangent, sine, and the Pythagorean theorem. It emphasizes drawing a diagram, identifying if it is an angle of elevation or depression, and using the appropriate trigonometric ratio to solve for missing lengths.
Adding and subtracting rational expressionsDawn Adams2
Using rules for fractions, rational expressions can be added and subtracted by finding common denominators. To find the common denominator, we find the least common multiple (LCM) of the denominators. With polynomials, the LCM will contain all factors of each denominator. We can then convert the fractions to equivalent forms using the LCM as the new denominator before combining like terms to evaluate the expression. Special cases may involve fractions with understood denominators of 1 or similar but non-equal denominators that can be made equal through factoring.
This document provides guidance on using a strategic intervention material (SIM) to learn about combined variation. It begins with reminders on using the SIM independently and thoroughly. The least mastered skill is solving problems involving combined variation, where a variable depends on two or more other variables in a direct and inverse relationship. Examples are provided of writing equations to represent relationships like "W varies jointly as c and the square of a and inversely as b." Several practice activities involve writing equations and solving for unknowns. The document concludes with an assessment to check understanding of combined variation concepts.
Lesson 1.9 b multiplication and division of rational numbersJohnnyBallecer
The document provides steps for multiplying rational numbers:
1) Write rational numbers in the form a/b ∙ c/d
2) Multiply the numerators and denominators separately
3) Simplify the resulting fraction by reducing if possible
The examples demonstrate changing mixed numbers to improper fractions, multiplying, and simplifying the final answer.
- The document discusses solving absolute value equations and inequalities.
- Absolute value equations will have two solutions, which are found by setting the expression inside the absolute value signs equal to the positive and negative of the right side of the equation.
- Absolute value inequalities require graphing the solutions on a number line. If the sign is >, the solutions are to the right. If <, the solutions are to the left.
- A multi-step example of solving an absolute value inequality is worked through.
This document discusses combined variation and how to solve problems involving quantities that vary directly and inversely with other variables. It provides examples of translating statements of combined variation into mathematical equations. It also works through an example problem, showing how to solve for an unknown variable value when the quantities it varies with are given. The document concludes by instructing the reader to practice additional combined variation problems from their workbook.
This document provides information about linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It discusses using the rectangular coordinate system to graph linear equations by plotting the x- and y-intercepts. It also describes how to determine if an ordered pair is a solution to a linear equation by substituting the x- and y-values into the equation. Finally, it briefly outlines common methods for solving systems of linear equations, including elimination, substitution, and cross-multiplication.
Finding Slope Given A Graph And Two PointsGillian Guiang
The student will learn to find the slope of a line given two points on a graph or explicitly given the points. Slope is defined as the steepness or rise over run of a line between two points. You can find the slope by taking the difference in the y-values and dividing by the difference in the x-values of the two points. Slope can be positive, negative, zero if horizontal, or undefined if vertical. Examples are worked through of finding the slope given two points on a graph or the points explicitly.
The document discusses slopes of lines. It defines slope as the ratio of the rise over the run between two points on a line. Specifically, if the points are (x1, y1) and (x2, y2), then the slope m is defined as (y2 - y1) / (x2 - x1). This ratio represents the steepness of the line, with larger absolute values indicating a steeper slope. An example calculates the slope of a line passing through the points (3, -2) and (-2, 8), finding the slope to be -2.
The document discusses slopes and how they are measured and calculated. It defines slope as the ratio of the rise over the run between two points on a line. The slope of a line can be calculated using the formula: m = (y2-y1)/(x2-x1). This measures the steepness of the line. Examples are given to demonstrate calculating the slope between two points and interpreting it geometrically as the ratio of rise over run. Slopes apply to lines, streets, roofs, and any surface with an incline or grade.
This document discusses integers and absolute value. It defines integers as positive and negative whole numbers and explains how to graph and order integers on a number line. It also defines absolute value as the distance from zero and how to evaluate the absolute value of integers, including that the absolute value of a number can never be negative. Examples are provided for graphing, ordering, and finding opposites and absolute values of integers.
This document discusses how to multiply and divide rational algebraic expressions. It explains that to multiply rational expressions, one multiplies the numerators and denominators separately. Rational expressions must first be factored to cancel common factors before multiplying. Several examples of multiplying rational expressions are shown. The document also explains that to divide rational expressions, one multiplies the numerator by the reciprocal of the denominator. Some examples of dividing rational expressions are provided.
Sample space, events, outcomes, and experimentsChristian Costa
Probability is a branch of mathematics that deals with describing how likely events are to occur. An experiment is a process with uncertain outcomes, while an outcome is any possible result. The sample space is the set of all possible outcomes. Fundamental counting principles state that if one event can occur in m ways, a second in n ways, and a third in p ways, the total number of outcomes is m x n x p. Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
The document discusses factoring the difference of two squares through examples such as (x+5)(x-5)=x^2 - 25. It explains that to factor a difference of two squares, we write the expression as the difference of two terms squared, then group the terms with the same bases and opposite signs inside parentheses. Several practice problems are provided to reinforce this technique for factoring completely the difference of two squares.
An axiomatic system consists of:
1) Undefined terms which are described but not defined;
2) Defined terms which are defined using undefined terms;
3) Axioms or postulates which are considered true without proof;
4) Theorems which are proved using axioms, definitions and logic.
The document discusses various theorems and properties related to triangles, including:
1) The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2) The triangle inequality theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.
3) Properties relating the lengths of sides and measures of angles in a triangle, such as if sides are unequal then angles will be unequal as well.
The document is a lecture on similar triangles. It defines similar triangles as having the same shape but different sizes, and discusses how similar triangles have corresponding angles that are congruent and corresponding sides that are proportional. It provides examples of similar triangles and statements showing their similarity. It also covers using proportions of corresponding sides to solve for missing sides in similar triangles and several proportionality principles related to similar triangles, including the basic proportionality theorem involving parallel lines cutting across a triangle.
This document defines slope and provides examples for teaching students about slope. It explains that slope is the ratio of vertical to horizontal change and can be positive, negative, zero, or undefined. The objectives are for students to identify slope from graphs, calculate slope using rise over run, and apply the slope formula to find slope given two points. Examples are provided to demonstrate calculating slope from graphs and points using rise over run and the slope formula.
This document explains the fundamental counting principle and permutations. It provides examples of how to use the principle to calculate the number of possible outcomes for events with multiple steps or choices. These include examples like counting meal combinations, sandwich varieties, possible license plates, phone numbers, test answers, and card selections. It discusses accounting for situations where items can or cannot be repeated between choices.
This document discusses finding the sum and product of the roots of quadratic equations. It provides the formulas for calculating the sum and product of roots without explicitly solving for the individual roots. The sum of the roots is equal to -b/a, and the product of the roots is equal to c/a, where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. Several examples are worked out applying these formulas. The document also contains exercises asking the reader to find the sum and product of roots for additional quadratic equations using the given coefficients a, b, and c.
Direct variation describes the relationship between two quantities where one quantity varies as the other changes proportionally. It can be represented by the equation y = kx, where k is the constant of variation.
Some key points about direct variation:
- The graph of a direct variation will pass through the origin, as there is no y-intercept term.
- To determine if a relationship represents direct variation, calculate the constant of variation k from the data and check if it remains the same for different values.
- Direct variation can be used to find unknown values by setting up a table with the known values and using the direct variation equation y = kx.
This document provides an introduction to basic probability concepts. It defines key terms like experiments, outcomes, sample space, and events. It explains how to calculate probabilities using fractions, decimals, or percentages. Examples are provided on determining the probability of rolling certain numbers on a die or spinning to certain areas on a spinner. The document also covers concepts like finding the probability of compound events and solving word problems involving probabilities.
12. Angle of Elevation & Depression.pptxBebeannBuar1
This document discusses angles of elevation and depression. It defines an angle of elevation as the angle formed between a horizontal line and the line of sight to an object located above the horizontal line. An angle of depression is defined as the angle formed between a horizontal line and the line of sight to an object located below the horizontal line. The document provides examples of solving problems involving angles of elevation and depression using trigonometric functions like tangent, sine, and the Pythagorean theorem. It emphasizes drawing a diagram, identifying if it is an angle of elevation or depression, and using the appropriate trigonometric ratio to solve for missing lengths.
Adding and subtracting rational expressionsDawn Adams2
Using rules for fractions, rational expressions can be added and subtracted by finding common denominators. To find the common denominator, we find the least common multiple (LCM) of the denominators. With polynomials, the LCM will contain all factors of each denominator. We can then convert the fractions to equivalent forms using the LCM as the new denominator before combining like terms to evaluate the expression. Special cases may involve fractions with understood denominators of 1 or similar but non-equal denominators that can be made equal through factoring.
This document provides guidance on using a strategic intervention material (SIM) to learn about combined variation. It begins with reminders on using the SIM independently and thoroughly. The least mastered skill is solving problems involving combined variation, where a variable depends on two or more other variables in a direct and inverse relationship. Examples are provided of writing equations to represent relationships like "W varies jointly as c and the square of a and inversely as b." Several practice activities involve writing equations and solving for unknowns. The document concludes with an assessment to check understanding of combined variation concepts.
Lesson 1.9 b multiplication and division of rational numbersJohnnyBallecer
The document provides steps for multiplying rational numbers:
1) Write rational numbers in the form a/b ∙ c/d
2) Multiply the numerators and denominators separately
3) Simplify the resulting fraction by reducing if possible
The examples demonstrate changing mixed numbers to improper fractions, multiplying, and simplifying the final answer.
- The document discusses solving absolute value equations and inequalities.
- Absolute value equations will have two solutions, which are found by setting the expression inside the absolute value signs equal to the positive and negative of the right side of the equation.
- Absolute value inequalities require graphing the solutions on a number line. If the sign is >, the solutions are to the right. If <, the solutions are to the left.
- A multi-step example of solving an absolute value inequality is worked through.
This document discusses combined variation and how to solve problems involving quantities that vary directly and inversely with other variables. It provides examples of translating statements of combined variation into mathematical equations. It also works through an example problem, showing how to solve for an unknown variable value when the quantities it varies with are given. The document concludes by instructing the reader to practice additional combined variation problems from their workbook.
This document provides information about linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It discusses using the rectangular coordinate system to graph linear equations by plotting the x- and y-intercepts. It also describes how to determine if an ordered pair is a solution to a linear equation by substituting the x- and y-values into the equation. Finally, it briefly outlines common methods for solving systems of linear equations, including elimination, substitution, and cross-multiplication.
Finding Slope Given A Graph And Two PointsGillian Guiang
The student will learn to find the slope of a line given two points on a graph or explicitly given the points. Slope is defined as the steepness or rise over run of a line between two points. You can find the slope by taking the difference in the y-values and dividing by the difference in the x-values of the two points. Slope can be positive, negative, zero if horizontal, or undefined if vertical. Examples are worked through of finding the slope given two points on a graph or the points explicitly.
The document discusses slopes of lines. It defines slope as the ratio of the rise over the run between two points on a line. Specifically, if the points are (x1, y1) and (x2, y2), then the slope m is defined as (y2 - y1) / (x2 - x1). This ratio represents the steepness of the line, with larger absolute values indicating a steeper slope. An example calculates the slope of a line passing through the points (3, -2) and (-2, 8), finding the slope to be -2.
The document discusses slopes and how they are measured and calculated. It defines slope as the ratio of the rise over the run between two points on a line. The slope of a line can be calculated using the formula: m = (y2-y1)/(x2-x1). This measures the steepness of the line. Examples are given to demonstrate calculating the slope between two points and interpreting it geometrically as the ratio of rise over run. Slopes apply to lines, streets, roofs, and any surface with an incline or grade.
The document defines slope as the ratio of the rise (change in y-values) to the run (change in x-values) between two points on a line. It provides the exact formula for calculating slope as the change in y-values divided by the change in x-values. Examples are given to demonstrate calculating the slopes of various lines, with positive slopes for lines passing through Quadrants I and III and negative slopes for lines passing through Quadrants II and IV.
The document defines slope as the ratio of the rise (change in y-values) to the run (change in x-values) between two points on a line. It provides the exact formula for calculating slope and gives examples of finding the slopes of various lines. Key points made include that horizontal lines have a slope of 0, vertical lines have an undefined slope, and lines through quadrants I and III have positive slopes while lines through quadrants II and IV have negative slopes.
The document defines slope as the ratio of the "rise" over the "run" between two points on a line. Specifically, if the points are (x1, y1) and (x2, y2), then the slope m is equal to (y2 - y1) / (x2 - x1). It also discusses how to calculate the slope of a line given two points, and how the slope indicates whether a line rises or falls from left to right. Lines between the first and third quadrants have positive slopes, while lines between the second and fourth quadrants have negative slopes.
The document discusses linear equations and graphing linear relationships. It defines key terms like slope, y-intercept, x-intercept, and provides examples of writing linear equations in slope-intercept form and point-slope form given certain information like a point and slope. It also discusses using tables and graphs to plot linear equations and find slopes from graphs or between two points. Real-world examples of linear relationships are provided as well.
The document provides information about linear equations and their graphs. It defines linear equations and discusses how to write equations in slope-intercept form, point-slope form, and standard form. It also describes how to graph linear equations by plotting intercepts and using slope. Key topics covered include finding the slope between two points, determining if lines are parallel or perpendicular based on their slopes, and recognizing the intercepts on a graph of a linear equation in two variables.
This document discusses different types of straight lines and their equations in a plane. It covers lines parallel to coordinate axes, lines through a given point with a given slope, lines with a given slope and y-intercept, and lines passing through two given points. It also discusses finding the equation of the line perpendicular to a given line and passing through a given point, as well as finding the distance from a point to a line. Examples of finding various line equations are provided.
This document contains information about linear functions and equations including:
- The domain and range of relations are specified using set builder notation.
- Linear functions are defined as having terms that are either constants or the product of a constant and a single variable to the first power.
- Slope-intercept form and standard form of linear equations are explained.
- Finding the x- and y- intercepts of a line from its equation is demonstrated.
- Slope is defined and calculations of slope between two points is shown.
- Lines are classified by their slopes and graphs of equations specifying x or y are given.
- Finding parallel and perpendicular lines from a given line is illustrated.
This document provides information about graphing linear equations. It begins by defining a linear equation as one whose solutions fall on a straight line. It explains how to identify if an equation is linear based on whether a constant change in the x-value corresponds to a constant change in the y-value. The document then gives examples of graphing equations and determining if they are linear based on whether their graphs form a straight line. It also discusses using tables to list the x and y-values of points that satisfy the equation.
This document discusses how to calculate the slope of a line from two points on the line. It provides the formula for slope as the rise over the run, or (y2 - y1) / (x2 - x1). Several examples are worked through, finding the slopes of lines passing through different pairs of points. The key points are:
- Slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points
- The formula to calculate slope from two points (x1, y1) and (x2, y2) is: (y2 - y1) / (x2 - x1)
- Slope can be positive, negative, zero
This document discusses how to calculate the slope of a line from two points on the line. It provides the formula for slope as the rise over the run, or (y2 - y1) / (x2 - x1). Several examples are worked through, finding the slopes of lines passing through different pairs of points. The key points are:
- Slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points
- The formula to calculate slope from two points (x1, y1) and (x2, y2) is: (y2 - y1) / (x2 - x1)
- Slope can be positive, negative, zero
The document discusses key concepts about linear equations in two variables including:
1) It describes the Cartesian coordinate plane and how to plot points based on their x and y coordinates.
2) It explains how to find the slope, y-intercept, and x-intercept of a linear equation graphically and algebraically.
3) It provides examples of rewriting linear equations in slope-intercept form (y=mx+b) and using intercepts and slopes to graph lines on the coordinate plane.
This document discusses linear equations and functions. It defines the slope-intercept form of a linear equation as y = mx + b, where m is the slope and b is the y-intercept. It explains how to determine the points on a line by substituting values for x and how properties of linear functions include having a constant slope and unlimited domain and range. The document also covers solving systems of linear equations by finding the point where two lines intersect.
Here are the key steps to find the instantaneous rate of change using a graphing calculator:
1. Graph the function over the appropriate domain.
2. Use the arrow keys to move the cursor to the point where you want to find the instantaneous rate of change.
3. Press the TRACE button and select the tangent option.
4. The calculator will display the slope of the tangent line, which is the instantaneous rate of change at that point.
5. For example, if finding the IROC at x=1 for the function f(x) = x3, you would:
a) Graph f(x) = x3
b) Use arrows to move cursor
The document discusses various topics in three dimensional geometry, including:
1. Coordinate systems define three perpendicular coordinate axes dividing space into eight parts.
2. The distance between two points P and Q is calculated as the square root of the sum of the squares of the differences between their x, y, and z coordinates.
3. Section formulae give the coordinates of points dividing or projecting onto lines between two given points.
- Derivatives describe how a quantity is changing with respect to something else, like how velocity changes over time.
- The derivative of a function y(x) at a point x is the slope of the tangent line to the curve of y(x) at that point.
- Mathematically, the derivative dy/dx is defined as the limit as h approaches 0 of the change in y over the change in x, (y(x+h)-y(x))/h.
- For functions of the form y(x)=Ax^n, the derivative has a shortcut of dy/dx=nAx^(n-1).
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their properties like slope and intercept points.
The document describes the rectangular coordinate system. It establishes that a coordinate system assigns positions in a plane using ordered pairs of numbers (x,y). It defines the x-axis, y-axis, and origin at their intersection. Any point is addressed by its coordinates (x,y) where x represents horizontal distance from the origin and y represents vertical distance. The four quadrants divided by the axes are also defined based on positive and negative coordinate values. Reflections of points across the axes and origin are discussed. Finally, it introduces the concept of graphing mathematical relations between x and y coordinates to represent collections of points.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
2 the real line, inequalities and comparative phraseselem-alg-sample
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the position of two numbers on the real number line, with the number farther to the right said to be greater than the number on the left. Examples are provided of drawing intervals on the number line and solving simple inequalities algebraically. Properties of inequalities like adding the same quantity to both sides preserving the inequality sign are also outlined.
Geometry is the study of shapes, their properties and relationships. Some basic geometric shapes include lines, rays, angles, triangles, quadrilaterals, polygons, circles and three-dimensional shapes like spheres and cubes. Formulas are used to calculate properties of shapes like the area of a triangle is 1/2 * base * height, the circumference of a circle is 2 * pi * radius, and the volume of a cube is side^3.
The document discusses direct and inverse variations. It defines a direct variation as a relationship where y=kx, where k is a constant. An inverse variation is defined as a relationship where y=k/x, where k is a constant. Examples are given of translating phrases describing direct and inverse variations into mathematical equations. The document also explains how to solve word problems involving variations by using given values to find the specific constant k and exact variation equation.
17 applications of proportions and the rational equationselem-alg-sample
The document discusses rational equations word problems involving rates, distances, costs, and number of people. An example problem asks how many people (x) shared a taxi costing $20 if one person leaving causes the remaining people's cost to increase by $1 each. Setting up rational equations and solving leads to the answer that x = 5 people.
16 the multiplier method for simplifying complex fractionselem-alg-sample
The document discusses two methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method reduces the complex fraction to an "easy" regular division problem by combining fractions in the numerator and denominator. The second method multiplies the lowest common denominator of all terms to the numerator and denominator to simplify. An example using each method is provided.
15 proportions and the multiplier method for solving rational equationselem-alg-sample
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding or subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate converting rational expressions to equivalent forms with different specified denominators.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
10 more on factoring trinomials and factoring by formulaselem-alg-sample
The document discusses two methods for factoring trinomials of the form ax^2 + bx + c. The first method is short but not always reliable, while the second method takes more steps but always provides a definite answer. This second method, called the reversed FOIL method, involves finding four numbers that satisfy certain properties to factor the trinomial. An example is worked out step-by-step to demonstrate how to use the reversed FOIL method to factor the trinomial 3x^2 + 5x + 2.
Trinomials are polynomials of the form ax^2 + bx + c, where a, b, and c are numbers. To factor a trinomial, we write it as the product of two binomials (x + u)(x + v) where uv = c and u + v = b. For example, to factor x^2 + 5x + 6, we set uv = 6 and u + v = 5. The only possible values are u = 2 and v = 3, so x^2 + 5x + 6 = (x + 2)(x + 3). Similarly, to factor x^2 - 5x + 6, we set uv = 6 and u + v = -5,
The document discusses factoring quantities by finding common factors. It defines factoring as rewriting a quantity as a product in a nontrivial way. A quantity is prime if it cannot be written as a product other than 1 times the quantity. To factor completely means writing each factor as a product of prime numbers. Examples show finding common factors of quantities, the greatest common factor (GCF), and extracting common factors from sums and differences using the extraction law.
The document discusses methods for multiplying binomial expressions. A binomial is a two-term polynomial of the form ax + b, while a trinomial is a three-term polynomial of the form ax^2 + bx + c. The product of two binomials results in a trinomial. The FOIL method is introduced to multiply binomials, where the Front, Outer, Inner, and Last terms of each binomial are multiplied and combined. Expanding the product of a binomial and a binomial with a leading negative sign requires distributing the negative sign first before using FOIL.
The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial in a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It then presents and explains the following rules for exponents:
1) Multiplication Rule: ANAK = AN+K
2) Division Rule: AN/AK = AN-K
3) Power Rule: (AN)K = ANK
4) 0-Power Rule: A0 = 1
5) Negative Power Rule: A-K = 1/AK
It provides examples to illustrate how to apply each rule when simplifying expressions with exponents.
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides steps to take which include clearing fractions by multiplying both sides by the LCD, moving all terms except the variable of interest to one side of the equation, and then dividing both sides by the coefficient of the isolated variable term to solve for the variable. Examples are provided to demonstrate these steps, such as solving for x in (a + b)x = c by dividing both sides by (a + b).
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
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This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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2. Slopes of Lines
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
3. Slopes of Lines
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
4. Slopes of Lines
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
Definition of Slope
5. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
6. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
7. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
8. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
9. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1 = 5 – (–2) = 7
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
10. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4,
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
11. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4,
then Δ x = x2 – x1
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
12. Slopes of Lines
Definition of Slope
Notation: The Greek capital letter Δ (delta) in general
means “the difference” in mathematics.
Δy means the difference in the values of y’s, Δx means the
difference the values of x’s.
Example A. Let y1 = –2, y2 = 5,
then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4,
then Δ x = x2 – x1 = –4 – 7 = –11
The slope of a line is a number. The slope of a line
measures the amount of tilt, (inclination, steepness) of the
line against the x-axis.
Steep lines have slopes with large absolute value.
Gradual lines have slopes with small absolute value
14. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
Slopes of Lines
(x1, y1)
(x2, y2)
15. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
m =
Slopes of Lines
(x1, y1)
(x2, y2)
16. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
17. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
Geometry of Slope
(x1, y1)
(x2, y2)
18. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
19. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
20. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
21. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
22. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
23. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
24. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
geometric
meaning
25. Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
26. Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
27. Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
28. (–2 , 8)
( 3 , –2)
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
29. (–2 , 8)
( 3 , –2)
–5 , 10
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
30. Δy
(–2 , 8)
( 3 , –2)
–5 , 10
Δx
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
31. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
32. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
33. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
34. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
35. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
(–2, 5)
( 3, 5)
36. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
Δy
(–2, 5)
( 3, 5)
–5, 0
Δx
37. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
Δy
(–2, 5)
( 3, 5)
–5, 0
Δx
So the slope is
Δx
Δy
m =
38. Δy
Δx
(–2 , 8)
( 3 , –2)
–5 , 10
Δy
Δx
Hence the slope is
10
–5
Example B. Find the slope of the line that passes through
(3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m = = = –2
Example C. Find the slope of the line
that passes through (3, 5) and (-2, 5).
Draw the line.
Δy
(–2, 5)
( 3, 5)
–5, 0
Δx
So the slope is
Δx
Δy 0
–5
m = = = 0
39. As shown in example F, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
40. As shown in example F, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
41. As shown in example F, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
42. As shown in example F, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
43. As shown in example F, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
So the slope
Δx
Δy 3
0
m = =
44. As shown in example F, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
So the slope
Δx
Δy 3
0
m = =
is undefined!
45. As shown in example F, the slope of a horizontal line is 0,
i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes
through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)
(3, 2)
0, 3
Δx
So the slope
Δx
Δy 3
0
m = =
is undefined!
As shown in example G, the slope of a vertical line is
undefined.
48. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
(x1, y1)
(x2, y2)
More on Slopes
49. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
m =
(x1, y1)
(x2, y2)
More on Slopes
50. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
(x1, y1)
(x2, y2)
More on Slopes
51. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
Geometry of Slope
(x1, y1)
(x2, y2)
More on Slopes
52. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
More on Slopes
53. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
More on Slopes
54. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
More on Slopes
55. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
More on Slopes
56. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
More on Slopes
57. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
More on Slopes
58. Definition of Slope
Let (x1, y1) and (x2, y2) be two points on a line,
then the slope m of the line is
Δy
Δx
y2 – y1
x2 – x1
m = =
rise
run=
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope
Δy = y2 – y1 = the difference
in the heights of the points.
Δx = x2 – x1 = the difference
in the runs of the points.
Δy
Δx
=Therefore m is the ratio of the “rise” to the “run”.
m =
Δy
Δx
y2 – y1
x2 – x1
=
easy to
memorize
the exact
formula
geometric
meaning
More on Slopes
59. Example A. Find the slope of each of the following lines.
More on Slopes
60. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
More on Slopes
61. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
More on Slopes
62. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
More on Slopes
63. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
More on Slopes
m =
Δy
Δx
=
0
7
= 0
64. Example A. Find the slope of each of the following lines.
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
More on Slopes
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0
65. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
More on Slopes
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0
66. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
More on Slopes
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0
67. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
More on Slopes
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0
68. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
More on Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
= 0
69. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
More on Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0
70. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
More on Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0
71. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
Δy = 3 – 3 = 0
More on Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0
72. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
Δy = 3 – 3 = 0
Δx = 6 – (–1) = 7
More on Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0
73. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
Δy = 3 – 3 = 0
Δx = 6 – (–1) = 7
More on Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
m =
Δy
Δx
=
7
0
Horizontal line
Slope = 0
Tilted line
Slope = 0
= 0 (UDF)
74. Example A. Find the slope of each of the following lines.
Two points are
(–2, –4), (2, 3).
Δy = 3 – (–4) = 7
Δx = 2 – (–2) = 4
m =
Two points are
(–3, 1), (4, 1).
Δy = 1 – (1) = 0
Δx = 4 – (–3) = 7
Two points are
(–1, 3), (6, 3).
Δy = 3 – 3 = 0
Δx = 6 – (–1) = 7
More on Slopes
Δy
Δx
=
7
4
m =
Δy
Δx
=
0
7
m =
Δy
Δx
=
7
0
Horizontal line
Slope = 0
Vertical line
Slope is UDF
Tilted line
Slope = 0
= 0 (UDF)
75. Lines that go through the
quadrants I and III have
positive slopes.
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76. Lines that go through the
quadrants I and III have
positive slopes.
More on Slopes
III
III IV
77. Lines that go through the
quadrants I and III have
positive slopes.
Lines that go through the
quadrants II and IV have
negative slopes.
More on Slopes
III
III IV
78. Lines that go through the
quadrants I and III have
positive slopes.
Lines that go through the
quadrants II and IV have
negative slopes.
More on Slopes
III
III IV
III
III IV
79. Lines that go through the
quadrants I and III have
positive slopes.
Lines that go through the
quadrants II and IV have
negative slopes.
More on Slopes
The formula for slopes requires geometric information,
i.e. the positions of two points on the line.
III
III IV
III
III IV
80. Lines that go through the
quadrants I and III have
positive slopes.
Lines that go through the
quadrants II and IV have
negative slopes.
More on Slopes
The formula for slopes requires geometric information,
i.e. the positions of two points on the line.
However, if a line is given by its equation instead, we may
determine the slope from the equation directly.
III
III IV
III
III IV
81. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
More on Slopes
82. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
More on Slopes
83. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
84. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
a. 3x = –2y + 6
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
85. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
a. 3x = –2y + 6 solve for y
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
86. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
87. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
88. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
89. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
and the y-intercept is (0, 3).
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
90. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
and the y-intercept is (0, 3).
Set y = 0, we get the x-intercept
(2, 0).
91. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
and the y-intercept is (0, 3).
Set y = 0, we get the x-intercept
(2, 0). Use these points to draw
the line.
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
92. Given a linear equation in x and y, solve for the variable y if
possible, we get y = mx + b
the number m is the slope and b is the y-intercept.
This is called the slope intercept form and this can be done
only if the y-term is present.
More on Slopes
a. 3x = –2y + 6 solve for y
2y = –3x + 6
y =
2
–3 x + 3
Hence the slope m is –3/2
and the y-intercept is (0, 3).
Set y = 0, we get the x-intercept
(2, 0). Use these points to draw
the line.
Example B. Write the equations into the slope intercept form,
list the slopes, the y-intercepts and draw the lines.
95. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
More on Slopes
96. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
More on Slopes
97. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
More on Slopes
98. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
More on Slopes
99. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
More on Slopes
100. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
More on Slopes
101. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
More on Slopes
102. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
More on Slopes
The variable y can’t be
isolated because there is no y.
103. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
More on Slopes
The variable y can’t be
isolated because there is no y.
Hence the slope is undefined
and this is a vertical line.
104. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
More on Slopes
The variable y can’t be
isolated because there is no y.
Hence the slope is undefined
and this is a vertical line.
Solve for x
3x = 6 x = 2.
105. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
More on Slopes
The variable y can’t be
isolated because there is no y.
Hence the slope is undefined
and this is a vertical line.
Solve for x
3x = 6 x = 2.
This is the vertical line x = 2.
106. b. 0 = –2y + 6 solve for y
2y = 6
y = 3
y = 0x + 3
Hence the slope m is 0.
The y-intercept is (0, 3).
There is no x-intercept.
c. 3x = 6
More on Slopes
The variable y can’t be
isolated because there is no y.
Hence the slope is undefined
and this is a vertical line.
Solve for x
3x = 6 x = 2.
This is the vertical line x = 2.
107. Two Facts About Slopes
I. Parallel lines have the same slope.
More on Slopes
108. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
More on Slopes
109. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
More on Slopes
110. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
More on Slopes
111. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
More on Slopes
112. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
More on Slopes
113. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
More on Slopes
114. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
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115. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
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b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
116. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
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b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Solve for y to find the slope of 3x – 4 = 2y
117. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
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b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Solve for y to find the slope of 3x – 4 = 2y
x – 2 = y2
3
118. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
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b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Solve for y to find the slope of 3x – 4 = 2y
x – 2 = y
Hence the slope of 3x = 2y + 4 is .
2
3
2
3
119. Two Facts About Slopes
I. Parallel lines have the same slope.
II. Slopes of perpendicular lines are the negative reciprocal of
each other.
Example C.
a. The line L is parallel to 4x – 2y = 5, what is the slope of L?
Solve for y for 4x – 2y = 5
4x – 5 = 2y
2x – 5/2 = y
So the slope of 4x – 2y = 5 is 2.
Since L is parallel to it , so L has slope 2 also.
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b. What is the slope of L if L is perpendicular to 3x = 2y + 4?
Solve for y to find the slope of 3x – 4 = 2y
x – 2 = y
Hence the slope of 3x = 2y + 4 is .
So L has slope –2/3 since L is perpendicular to it.
2
3
2
3
120. Summary on Slopes
How to Find Slopes
I. If two points on the line are given, use the slope formula
II. If the equation of the line is given, solve for the y and get
slope intercept form y = mx + b, then the number m is
the slope.
Geometry of Slope
The slope of tilted lines are nonzero.
Lines with positive slopes connect quadrants I and III.
Lines with negative slopes connect quadrants II and IV.
Lines that have slopes with large absolute values are steep.
The slope of a horizontal line is 0.
A vertical lines does not have slope or that it’s UDF.
Parallel lines have the same slopes.
Perpendicular lines have the negative reciprocal slopes of
each other.
rise
run=m =
Δy
Δx
y2 – y1
x2 – x1
=
121. Exercise A. Identify the vertical and the horizontal lines by
inspection first. Find their slopes or if it’s undefined, state so.
Fine the slopes of the other ones by solving for the y.
1. x – y = 3 2. 2x = 6 3. –y – 7= 0
4. 0 = 8 – 2x 5. y = –x + 4 6. 2x/3 – 3 = 6/5
7. 2x = 6 – 2y 8. 4y/5 – 12 = 3x/4 9. 2x + 3y = 3
10. –6 = 3x – 2y 11. 3x + 2 = 4y + 3x 12. 5x/4 + 2y/3 = 2
Exercise B.
13–18. Select two points and estimate the slope of each line.
13. 14. 15.
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122. 16. 17. 18.
Exercise C. Draw and find the slope of the line that passes
through the given two points. Identify the vertical line and the
horizontal lines by inspection first.
19. (0, –1), (–2, 1) 20. (1, –2), (–2, 0) 21. (1, –2), (–2, –1)
22. (3, –1), (3, 1) 23. (1, –2), (–2, 3) 24. (2, –1), (3, –1)
25. (4, –2), (–3, 1) 26. (4, –2), (4, 0) 27. (7, –2), (–2, –6)
28. (3/2, –1), (3/2, 1) 29. (3/2, –1), (1, –3/2)
30. (–5/2, –1/2), (1/2, 1) 31. (3/2, 1/3), (1/3, 1/3)
32. (–2/3, –1/4), (1/2, 2/3) 33. (3/4, –1/3), (1/3, 3/2)
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123. Exercise D.
34. Identify which lines are parallel and which one are
perpendicular.
A. The line that passes through (0, 1), (1, –2)
D. 2x – 4y = 1
B. C.
E. The line that’s perpendicular to 3y = x
F. The line with the x–intercept at 3 and y intercept at 6.
Find the slope, if possible of each of the following lines.
35. The line passes with the x intercept at x = 2,
and y–intercept at y = –5.
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124. 36. The equation of the line is 3x = –5y+7
37. The equation of the line is 0 = –5y+7
38. The equation of the line is 3x = 7
39. The line is parallel to 2y = 5 – 6x
40. the line is perpendicular to 2y = 5 – 6x
41. The line is parallel to the line in problem 30.
42. the line is perpendicular to line in problem 31.
43. The line is parallel to the line in problem 33.
44. the line is perpendicular to line in problem 34.
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Find the slope, if possible of each of the following lines
125. Summary of Slope
The slope of the line that passes through (x1, y1) and (x2, y2) is
Horizontal line
Slope = 0
Vertical line
Slope is UDF.
Tilted line
Slope = –2 0
rise
run
=m =
Δy
Δx
y2 – y1
x2 – x1
=
126. Exercise A.
Select two points and estimate the slope of each line.
1. 2. 3. 4.
Slopes of Lines
5. 6. 7. 8.
127. Exercise B. Draw and find the slope of the line that passes
through the given two points. Identify the vertical line and the
horizontal lines by inspection first.
9. (0, –1), (–2, 1) 10. (1, –2), (–2, 0) 11. (1, –2), (–2, –1)
12. (3, –1), (3, 1) 13. (1, –2), (–2, 3) 14. (2, –1), (3, –1)
15. (4, –2), (–3, 1) 16. (4, –2), (4, 0) 17. (7, –2), (–2, –6)
18. (3/2, –1), (3/2, 1) 19. (3/2, –1), (1, –3/2)
20. (–5/2, –1/2), (1/2, 1) 21. (3/2, 1/3), (1/3, 1/3)
22. (–2/3, –1/4), (1/2, 2/3) 23. (3/4, –1/3), (1/3, 3/2)
Slopes of Lines