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.
This document provides instruction on using present forms of verbs in English sentences. It discusses using the simple form with plural subjects and the "s" form with singular subjects. Examples are given of community workers and sentences about them. Students are asked to analyze sentences about workers to identify the verb forms used with singular and plural subjects. They then practice completing sentences and poems by choosing the correct verb form.
English 6 dlp 5 words with affixes - prefixes optEDITHA HONRADEZ
The document discusses prefixes and how they can be used to form new words. It provides examples of common prefixes like "un-", "in-", "dis-", "im-", and "ir-" which are often used to mean "opposite of" or "not". Learners are given exercises to practice identifying prefixes in words and using prefixes to complete sentences. The purpose is to help expand one's vocabulary through understanding and using prefixes.
This document outlines a proposed curriculum for a spelling course for 10th grade students in Nizwa Secondary School. It includes a needs analysis showing students have difficulty with spelling skills. The course aims to improve spelling abilities through strategies like looking-thinking-covering-writing words. It is scheduled over 15 weeks and includes vocabulary lists, dictation quizzes, writing assignments, and games. Assessment includes dictation tests, writing tests, a reading test, and a final exam. Upon completion, students will reflect on their learning and the course's effectiveness.
This document is a module from an English class that teaches students about analyzing word structures and meanings. It includes the following:
- An introduction to identifying meanings of unfamiliar words through analyzing prefixes and suffixes.
- Examples of words with prefixes like "ir-", "im-", "dis-", "sub-", "over-", and "semi-" and their meanings.
- Examples of words with suffixes like "-ful", "-less", "-er", "-ant", "-able", and "-ery" and their meanings.
- Exercises for students to practice identifying root words and meanings of words with certain prefixes and suffixes.
- Instructions for students to complete a performance task as an assignment analyzing additional words.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
This document provides instruction on using present forms of verbs in English sentences. It discusses using the simple form with plural subjects and the "s" form with singular subjects. Examples are given of community workers and sentences about them. Students are asked to analyze sentences about workers to identify the verb forms used with singular and plural subjects. They then practice completing sentences and poems by choosing the correct verb form.
English 6 dlp 5 words with affixes - prefixes optEDITHA HONRADEZ
The document discusses prefixes and how they can be used to form new words. It provides examples of common prefixes like "un-", "in-", "dis-", "im-", and "ir-" which are often used to mean "opposite of" or "not". Learners are given exercises to practice identifying prefixes in words and using prefixes to complete sentences. The purpose is to help expand one's vocabulary through understanding and using prefixes.
This document outlines a proposed curriculum for a spelling course for 10th grade students in Nizwa Secondary School. It includes a needs analysis showing students have difficulty with spelling skills. The course aims to improve spelling abilities through strategies like looking-thinking-covering-writing words. It is scheduled over 15 weeks and includes vocabulary lists, dictation quizzes, writing assignments, and games. Assessment includes dictation tests, writing tests, a reading test, and a final exam. Upon completion, students will reflect on their learning and the course's effectiveness.
This document is a module from an English class that teaches students about analyzing word structures and meanings. It includes the following:
- An introduction to identifying meanings of unfamiliar words through analyzing prefixes and suffixes.
- Examples of words with prefixes like "ir-", "im-", "dis-", "sub-", "over-", and "semi-" and their meanings.
- Examples of words with suffixes like "-ful", "-less", "-er", "-ant", "-able", and "-ery" and their meanings.
- Exercises for students to practice identifying root words and meanings of words with certain prefixes and suffixes.
- Instructions for students to complete a performance task as an assignment analyzing additional words.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
teaching english in elementary grades through literature.pptxPascualJaniceC
Science, Technology & Society (STS) is an interdisciplinary field of study that seeks to explore and understand the many ways that modern science and technology shape modern culture, values, and institutions, and how modern values shape science and technology.
This document describes the function of each part of the respiratory system. It begins by listing the parts and their functions: nostrils allow air to enter the nasal cavity, the pharynx enables air to pass to the trachea, the larynx produces sounds, the trachea transports air to the lungs, bronchioles allow for gas exchange, and lungs filter and purify air. The document then provides a lesson plan to teach students about the respiratory system through activities, discussion of the parts and their roles, and a labeling exercise to reinforce understanding.
The document discusses filling out forms, with the objectives of identifying what forms are, learning how to fill them out correctly and honestly, and being able to do so for things like school forms. It states that forms are used to gather essential information and provide templates, with some requiring more details than others. Students are expected to identify common forms they use, fill them out accurately and legibly, and recognize the importance of providing correct information.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined, and tilted lines. Horizontal lines have the equation y=c, vertical lines x=c, and tilted lines are found using the point-slope formula y=m(x-x1)+y1, where m is the slope and (x1,y1) is a point on the line. Examples are given to demonstrate finding equations of lines given information about them.
This document provides a lesson on identifying and writing equations for parallel and perpendicular lines. It includes examples of identifying parallel lines based on their having the same slope. Perpendicular lines are defined as having slopes whose product is -1. The document also demonstrates how to write equations for lines parallel or perpendicular to a given line based on their point and slope. Examples are provided to illustrate finding slopes from graphs and using them to determine if lines are parallel or perpendicular.
This document contains information about straight lines and their various forms of equations. It discusses the slope-intercept form, intercept form, two-point form and general form of a straight line. It provides examples of finding the equation of a line given certain conditions like two points, slope and intercept, perpendicular or parallel lines etc. There are also examples of problems involving finding slope, intercepts, perpendicular or parallel lines to a given line. The document contains the necessary formulas and steps to solve such problems.
This document provides examples and explanations about parallel and perpendicular lines. It defines parallel lines as lines in the same plane that do not intersect. Perpendicular lines intersect to form a 90 degree angle. Several examples show how to identify parallel and perpendicular lines based on their slopes. The document also demonstrates applications to geometry, such as showing when a figure like a parallelogram or right triangle is formed based on the relationships between line slopes.
The document discusses parallel and perpendicular lines. It provides examples of identifying parallel and perpendicular lines based on their slopes. It also discusses writing equations of lines parallel and perpendicular to given lines, and applications involving parallel and perpendicular lines in geometry problems involving shapes like parallelograms, rectangles, and right triangles.
The document discusses point-slope form and writing linear equations. It provides examples of writing equations for lines that are parallel and perpendicular to given lines. Specifically, it explains that parallel lines have the same slope and different y-intercepts, while perpendicular lines have slopes that are negative reciprocals of each other. Examples are given of writing equations for parallel and perpendicular lines using point-slope form based on a given point and the slope of another line.
The document discusses the concept of slope and how it is used to describe the steepness of a line. It defines slope as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Several forms of linear equations are presented, including point-slope form, slope-intercept form, and standard form. Relationships between parallel and perpendicular lines based on their slopes are also described. Examples are provided to demonstrate finding slopes, writing equations of lines, and determining if lines are parallel or perpendicular based on their slopes.
This lecture discusses distance, midpoint, slope, lines, symmetries of graphs, equations of circles, and quadratic equations. It defines distance as the square root of the sum of the squared differences of x- and y-coordinates between two points. The midpoint formula finds the midpoint of a line segment between two points. Slope is defined as the rise over the run between two points on a line. Lines can be written in point-slope form, slope-intercept form, and intercept forms. Parallel and perpendicular lines are identified based on equal or negative reciprocal slopes. Symmetries of graphs include reflections across the x-axis, y-axis, or origin. The equation of a circle is given by (x-h)2
The document provides information about curve tracing including important definitions, the method of tracing a curve, and examples of tracing specific curves. It defines singular points, multiple points, nodes, cusps, and points of inflection. The method of tracing involves analyzing the curve for symmetry, points of intersection with the axes, regions where the curve does not exist, asymptotes, and tangents. Examples analyze the curves y=(x-a)^2, (x+y)^2=(x-a)^2, y=(2-x)^2, and y=x^2 for these properties and sketch the curves.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Lecture 15(graphing of cartesion curves)FahadYaqoob5
1. The curve passes through the origin as there is no constant term in the equation. To find the tangents at the origin, the lowest degree terms are equated to zero, giving the x-axis as a tangent.
2. The curve meets the coordinate axes only at the origin. There are no other points of intersection.
3. The curve has no asymptotes parallel to the x-axis, but the line y=2a is an asymptote parallel to the y-axis. The curve exists only in the region 0<x<2a.
1. The curve passes through the origin as there is no constant term in the equation. To find the tangents at the origin, the lowest degree terms are equated to zero, giving the x-axis as a tangent.
2. The curve meets the coordinate axes only at the origin. There are no other points of intersection.
3. The curve is symmetric about the x-axis as the powers of y in the equation are even. There are no other symmetries.
4. The y-axis is the only asymptote, obtained by equating the coefficient of the highest power of y to zero. There is no asymptote parallel to the x-axis.
5. The curve exists in
1) The document defines algebraic curves as geometric figures formed by the set of points satisfying a given equation relating x and y.
2) Key properties of algebraic curves include their extent (domain and range), symmetry, intercepts, and asymptotes. Symmetry can be tested by substituting -x or -y. Intercepts are where the curve crosses the axes. Asymptotes are lines a curve approaches but never touches.
3) Tracing a curve involves determining its region, testing for symmetry, finding intercepts and asymptotes, and plotting points to sketch the curve.
Area Under Curves Basic Concepts - JEE Main 2015 Ednexa
This document outlines the key steps for curve sketching:
(1) Determine any symmetries based on even/odd powers of x and y. Symmetries may exist about the x-axis, y-axis, in opposite quadrants, or about the line y=x.
(2) Check if the curve passes through the origin by looking for constant terms.
(3) Find the points of intersection with the x-axis and y-axis by setting y=0 and x=0.
(4) Identify any special points where the tangent is parallel to the x-axis or y-axis.
(5) Determine the region bounded by the minimum and maximum x-values
teaching english in elementary grades through literature.pptxPascualJaniceC
Science, Technology & Society (STS) is an interdisciplinary field of study that seeks to explore and understand the many ways that modern science and technology shape modern culture, values, and institutions, and how modern values shape science and technology.
This document describes the function of each part of the respiratory system. It begins by listing the parts and their functions: nostrils allow air to enter the nasal cavity, the pharynx enables air to pass to the trachea, the larynx produces sounds, the trachea transports air to the lungs, bronchioles allow for gas exchange, and lungs filter and purify air. The document then provides a lesson plan to teach students about the respiratory system through activities, discussion of the parts and their roles, and a labeling exercise to reinforce understanding.
The document discusses filling out forms, with the objectives of identifying what forms are, learning how to fill them out correctly and honestly, and being able to do so for things like school forms. It states that forms are used to gather essential information and provide templates, with some requiring more details than others. Students are expected to identify common forms they use, fill them out accurately and legibly, and recognize the importance of providing correct information.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined, and tilted lines. Horizontal lines have the equation y=c, vertical lines x=c, and tilted lines are found using the point-slope formula y=m(x-x1)+y1, where m is the slope and (x1,y1) is a point on the line. Examples are given to demonstrate finding equations of lines given information about them.
This document provides a lesson on identifying and writing equations for parallel and perpendicular lines. It includes examples of identifying parallel lines based on their having the same slope. Perpendicular lines are defined as having slopes whose product is -1. The document also demonstrates how to write equations for lines parallel or perpendicular to a given line based on their point and slope. Examples are provided to illustrate finding slopes from graphs and using them to determine if lines are parallel or perpendicular.
This document contains information about straight lines and their various forms of equations. It discusses the slope-intercept form, intercept form, two-point form and general form of a straight line. It provides examples of finding the equation of a line given certain conditions like two points, slope and intercept, perpendicular or parallel lines etc. There are also examples of problems involving finding slope, intercepts, perpendicular or parallel lines to a given line. The document contains the necessary formulas and steps to solve such problems.
This document provides examples and explanations about parallel and perpendicular lines. It defines parallel lines as lines in the same plane that do not intersect. Perpendicular lines intersect to form a 90 degree angle. Several examples show how to identify parallel and perpendicular lines based on their slopes. The document also demonstrates applications to geometry, such as showing when a figure like a parallelogram or right triangle is formed based on the relationships between line slopes.
The document discusses parallel and perpendicular lines. It provides examples of identifying parallel and perpendicular lines based on their slopes. It also discusses writing equations of lines parallel and perpendicular to given lines, and applications involving parallel and perpendicular lines in geometry problems involving shapes like parallelograms, rectangles, and right triangles.
The document discusses point-slope form and writing linear equations. It provides examples of writing equations for lines that are parallel and perpendicular to given lines. Specifically, it explains that parallel lines have the same slope and different y-intercepts, while perpendicular lines have slopes that are negative reciprocals of each other. Examples are given of writing equations for parallel and perpendicular lines using point-slope form based on a given point and the slope of another line.
The document discusses the concept of slope and how it is used to describe the steepness of a line. It defines slope as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Several forms of linear equations are presented, including point-slope form, slope-intercept form, and standard form. Relationships between parallel and perpendicular lines based on their slopes are also described. Examples are provided to demonstrate finding slopes, writing equations of lines, and determining if lines are parallel or perpendicular based on their slopes.
This lecture discusses distance, midpoint, slope, lines, symmetries of graphs, equations of circles, and quadratic equations. It defines distance as the square root of the sum of the squared differences of x- and y-coordinates between two points. The midpoint formula finds the midpoint of a line segment between two points. Slope is defined as the rise over the run between two points on a line. Lines can be written in point-slope form, slope-intercept form, and intercept forms. Parallel and perpendicular lines are identified based on equal or negative reciprocal slopes. Symmetries of graphs include reflections across the x-axis, y-axis, or origin. The equation of a circle is given by (x-h)2
The document provides information about curve tracing including important definitions, the method of tracing a curve, and examples of tracing specific curves. It defines singular points, multiple points, nodes, cusps, and points of inflection. The method of tracing involves analyzing the curve for symmetry, points of intersection with the axes, regions where the curve does not exist, asymptotes, and tangents. Examples analyze the curves y=(x-a)^2, (x+y)^2=(x-a)^2, y=(2-x)^2, and y=x^2 for these properties and sketch the curves.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Lecture 15(graphing of cartesion curves)FahadYaqoob5
1. The curve passes through the origin as there is no constant term in the equation. To find the tangents at the origin, the lowest degree terms are equated to zero, giving the x-axis as a tangent.
2. The curve meets the coordinate axes only at the origin. There are no other points of intersection.
3. The curve has no asymptotes parallel to the x-axis, but the line y=2a is an asymptote parallel to the y-axis. The curve exists only in the region 0<x<2a.
1. The curve passes through the origin as there is no constant term in the equation. To find the tangents at the origin, the lowest degree terms are equated to zero, giving the x-axis as a tangent.
2. The curve meets the coordinate axes only at the origin. There are no other points of intersection.
3. The curve is symmetric about the x-axis as the powers of y in the equation are even. There are no other symmetries.
4. The y-axis is the only asymptote, obtained by equating the coefficient of the highest power of y to zero. There is no asymptote parallel to the x-axis.
5. The curve exists in
1) The document defines algebraic curves as geometric figures formed by the set of points satisfying a given equation relating x and y.
2) Key properties of algebraic curves include their extent (domain and range), symmetry, intercepts, and asymptotes. Symmetry can be tested by substituting -x or -y. Intercepts are where the curve crosses the axes. Asymptotes are lines a curve approaches but never touches.
3) Tracing a curve involves determining its region, testing for symmetry, finding intercepts and asymptotes, and plotting points to sketch the curve.
Area Under Curves Basic Concepts - JEE Main 2015 Ednexa
This document outlines the key steps for curve sketching:
(1) Determine any symmetries based on even/odd powers of x and y. Symmetries may exist about the x-axis, y-axis, in opposite quadrants, or about the line y=x.
(2) Check if the curve passes through the origin by looking for constant terms.
(3) Find the points of intersection with the x-axis and y-axis by setting y=0 and x=0.
(4) Identify any special points where the tangent is parallel to the x-axis or y-axis.
(5) Determine the region bounded by the minimum and maximum x-values
This document covers key concepts about lines in mathematics including:
- Lines have a constant slope which can be calculated using the slope formula.
- The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.
- Lines are parallel if they have the same slope and perpendicular if their slopes are negative reciprocals.
- Several examples are provided of finding the equation of a line given different conditions like two points or being parallel/perpendicular to another line.
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 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.
Similar to 6 equations and applications of lines (20)
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.
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).
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.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
1. Given enough information about a line, we can reconstruct an
equation of the line.
Equations of Lines
2. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
Equations of Lines
3. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
Case I. Horizontal and Vertical Lines (The Special Case)
Equations of Lines
4. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0.
Case I. Horizontal and Vertical Lines (The Special Case)
Equations of Lines
5. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
Equations of Lines
6. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
y= 3
Equations of Lines
Horizontal lines have slope 0.
7. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
y= 3
y=1½
Equations of Lines
Horizontal lines have slope 0.
8. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
y= –3
y= 3
y=1½
Equations of Lines
Horizontal lines have slope 0.
9. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
y= –3
y= 3
y=1½
Equations of Lines
Horizontal lines have slope 0.
The slope of vertical lines is
undefined, i.e. there is no “y” in
the equation.
10. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
y= –3
y= 3
y=1½
Equations of Lines
Horizontal lines have slope 0.
The slope of vertical lines is
undefined, i.e. there is no “y” in
the equation. So the equations
of vertical lines are x = c.
11. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
y= –3
y= 3
y=1½
Equations of Lines
Horizontal lines have slope 0.
The slope of vertical lines is
undefined, i.e. there is no “y” in
the equation. So the equations
of vertical lines are x = c.
Slope of vertical line is undefined.
x= 5
12. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
y= –3
y= 3
y=1½
Equations of Lines
Horizontal lines have slope 0.
The slope of vertical lines is
undefined, i.e. there is no “y” in
the equation. So the equations
of vertical lines are x = c.
Slope of vertical line is undefined.
x= 2 x= 5
13. Given enough information about a line, we can reconstruct an
equation of the line. We separate them into two cases.
The slope of horizontal
lines is 0. Hence the
equations of
horizontal lines are y = c.
Case I. Horizontal and Vertical Lines (The Special Case)
y= –3
y= 3
y=1½
Equations of Lines
Horizontal lines have slope 0.
The slope of vertical lines is
undefined, i.e. there is no “y” in
the equation. So the equations
of vertical lines are x = c.
Slope of vertical line is undefined.
x= –4 x= 2 x= 5
18. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line.
19. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line. So the
equation is x = c for some c.
20. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line. So the
equation is x = c for some c.
Since (3, –1) is on the line so
the equation must be x = 3.
21. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line. So the
equation is x = c for some c.
Since (3, –1) is on the line so
the equation must be x = 3.
b. A line passes through (3, –1 )
and it’s parallel to the x-axis.
Draw. Find its equation.
22. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line. So the
equation is x = c for some c.
Since (3, –1) is on the line so
the equation must be x = 3.
b. A line passes through (3, –1 )
and it’s parallel to the x-axis.
Draw. Find its equation.
23. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line. So the
equation is x = c for some c.
Since (3, –1) is on the line so
the equation must be x = 3.
b. A line passes through (3, –1 )
and it’s parallel to the x-axis.
Draw. Find its equation.
24. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line. So the
equation is x = c for some c.
Since (3, –1) is on the line so
the equation must be x = 3.
b. A line passes through (3, –1 )
and it’s parallel to the x-axis.
Draw. Find its equation.
Because it’s parallel to the x-
axis, it must be a horizontal
line.
25. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line. So the
equation is x = c for some c.
Since (3, –1) is on the line so
the equation must be x = 3.
b. A line passes through (3, –1 )
and it’s parallel to the x-axis.
Draw. Find its equation.
Because it’s parallel to the x-
axis, it must be a horizontal
line. So the equation is y = c
for some c.
26. Equations of Lines
Example A.
a. A line passes through (3, –1 ),
(3, –3). Draw. Find its equation.
It’s a vertical line. So the
equation is x = c for some c.
Since (3, –1) is on the line so
the equation must be x = 3.
b. A line passes through (3, –1 )
and it’s parallel to the x-axis.
Draw. Find its equation.
Because it’s parallel to the x-
axis, it must be a horizontal
line. So the equation is y = c
for some c. Since (3, –1) is on
the line so the equation must
be y = –1.
28. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
29. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly.
30. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
31. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
The Point Slope Formula (for composing the equations)
32. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line,
The Point Slope Formula (for composing the equations)
33. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line, then
The Point Slope Formula (for composing the equations)
y = m(x – x1) + y1
is the equation of the line.
34. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line, then
The Point Slope Formula (for composing the equations)
y = m(x – x1) + y1
is the equation of the line.
35. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line, then
The Point Slope Formula (for composing the equations)
y = m(x – x1) + y1
is the equation of the line.
Example B. Find the equations of the following lines.
a. The line with slope -2 and y-intercept at -7.
36. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line, then
The Point Slope Formula (for composing the equations)
y = m(x – x1) + y1
is the equation of the line.
The slope is –2, the point is (0, –7).
Example B. Find the equations of the following lines.
a. The line with slope -2 and y-intercept at -7.
37. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line, then
The Point Slope Formula (for composing the equations)
y = m(x – x1) + y1
is the equation of the line.
The slope is –2, the point is (0, –7). Hence,
y = –2(x
Example B. Find the equations of the following lines.
a. The line with slope -2 and y-intercept at -7.
38. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line, then
The Point Slope Formula (for composing the equations)
y = m(x – x1) + y1
is the equation of the line.
The slope is –2, the point is (0, –7). Hence,
y = –2(x – 0)
Example B. Find the equations of the following lines.
a. The line with slope -2 and y-intercept at -7.
39. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line, then
The Point Slope Formula (for composing the equations)
y = m(x – x1) + y1
is the equation of the line.
The slope is –2, the point is (0, –7). Hence,
y = –2(x – 0) + (–7)
Example B. Find the equations of the following lines.
a. The line with slope -2 and y-intercept at -7.
40. Equations of Lines
Case II. Tilted Lines (The General Case)
To find the equations of tilted lines, use the formula below.
It gives the slope-intercept equations directly. We need the
slope and a point on the line to use this formula.
Given the slope m, and a point (x1, y1) on the line, then
The Point Slope Formula (for composing the equations)
y = m(x – x1) + y1
is the equation of the line.
The slope is –2, the point is (0, –7). Hence,
y = –2(x – 0) + (–7)
or y = –2x – 7
Example B. Find the equations of the following lines.
a. The line with slope -2 and y-intercept at -7.
41. b. The line that contains (1, –2) with the x-intercept at –4.
Equations of Lines
42. b. The line that contains (1, –2) with the x-intercept at –4.
We have two points on the line (1, –2), (–4, 0) and we
need the slope.
Equations of Lines
43. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
m =
Equations of Lines
44. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
=
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
m =
Equations of Lines
45. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=m =
Equations of Lines
46. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
m =
Equations of Lines
y = m(x – x1) + y1
47. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
m =
Equations of Lines
y = m(x – x1) + y1
48. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
y =
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
5
–2
(x – (–4)) + 0
m =
Equations of Lines
y = m(x – x1) + y1
49. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
y =
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
5
–2
(x – (–4)) + 0
m =
Equations of Lines
y = m(x – x1) + y1
50. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
y =
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
5
–2
(x – (–4)) + 0
m =
Equations of Lines
y = m(x – x1) + y1
51. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
y =
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
5
–2
(x – (–4)) + 0
m =
Equations of Lines
52. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
y =
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
5
–2
(x – (–4)) + 0
m =
y = (x + 4)
5
–2
Equations of Lines
53. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
y =
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
5
–2
(x – (–4)) + 0
m =
y = (x + 4)
5
–2
y = x –
5
–2 8
5
Equations of Lines
54. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
y =
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
5
–2
(x – (–4)) + 0
m =
y = (x + 4)
5
–2
y = x –
5
–2 8
5
Equations of Lines
(or 5y = –2x – 8)
55. b. The line that contains (1, –2) with the x-intercept at –4.
Δy
Δx
0 – (–2 )
–4 – (1)
2
–5
=
y =
We have two points on the line (1, –2), (–4, 0) and we
need the slope. Use the slope formula,
=
using the point (–4, 0), plug in the Point Slope Formula
5
–2
(x – (–4)) + 0
m =
y = (x + 4)
5
–2
y = x –
5
–2 8
5
Equations of Lines
Recall that parallel lines have the same slope and
perpendicular lines have slopes that are the negative
reciprocals of each other.
(or 5y = –2x – 8)
56. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
Equations of Lines
57. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
58. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
To find the slope of 3y – 4x = 2, solve for the y.
59. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
To find the slope of 3y – 4x = 2, solve for the y.
3y = 4x + 2
60. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
To find the slope of 3y – 4x = 2, solve for the y.
3y = 4x + 2
y =
4
3 x + 2
3
61. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
4
3
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
To find the slope of 3y – 4x = 2, solve for the y.
Therefore the slope of the line 3y – 4x = 2 is .
3y = 4x + 2
y =
4
3 x + 2
3
62. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
4
3
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
To find the slope of 3y – 4x = 2, solve for the y.
Therefore the slope of the line 3y – 4x = 2 is .
3y = 4x + 2
y =
4
3 x + 2
3
So our line has slope .4
3
63. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
4
3
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
To find the slope of 3y – 4x = 2, solve for the y.
Therefore the slope of the line 3y – 4x = 2 is .
3y = 4x + 2
y =
4
3 x + 2
3
By the point-slope formula, the equation is
So our line has slope .4
3
64. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
4
3
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
To find the slope of 3y – 4x = 2, solve for the y.
Therefore the slope of the line 3y – 4x = 2 is .
3y = 4x + 2
y =
4
3 x + 2
3
y = (x – 3) + (–1)
By the point-slope formula, the equation is
So our line has slope .4
3
4
3
65. c. The line that passes through (3, –1) and is parallel to the
line 3y – 4x = 2.
4
3
Equations of Lines
Our line has the same slope as the line 3y – 4x = 2.
To find the slope of 3y – 4x = 2, solve for the y.
Therefore the slope of the line 3y – 4x = 2 is .
3y = 4x + 2
y =
4
3 x + 2
3
y = (x – 3) + (–1)
By the point-slope formula, the equation is
So our line has slope .4
3
4
3
y = 4
3
x – 4 – 1
66. d. The line that has y-intercept at –3 and is perpendicular to
the line 2x – 3y = 2.
Equations of Lines
67. d. The line that has y-intercept at –3 and is perpendicular to
the line 2x – 3y = 2.
Equations of Lines
For the slope, solve 2x – 3y = 2
68. d. The line that has y-intercept at –3 and is perpendicular to
the line 2x – 3y = 2.
Equations of Lines
For the slope, solve 2x – 3y = 2
–3y = –2x + 2
69. d. The line that has y-intercept at –3 and is perpendicular to
the line 2x – 3y = 2.
2
3
Equations of Lines
For the slope, solve 2x – 3y = 2
–3y = –2x + 2
y = 2
3
x –
70. d. The line that has y-intercept at –3 and is perpendicular to
the line 2x – 3y = 2.
2
3
Equations of Lines
For the slope, solve 2x – 3y = 2
–3y = –2x + 2
y = 2
3
x –
Hence the slope of 2x – 3y = 2 is .
2
3
71. d. The line that has y-intercept at –3 and is perpendicular to
the line 2x – 3y = 2.
2
3
Equations of Lines
For the slope, solve 2x – 3y = 2
–3y = –2x + 2
y =
Since perpendicular lines have slopes that are the
negative reciprocals of each other, our slope is .
2
3
x –
Hence the slope of 2x – 3y = 2 is .
2
3
–3
2
72. d. The line that has y-intercept at –3 and is perpendicular to
the line 2x – 3y = 2.
2
3
Equations of Lines
For the slope, solve 2x – 3y = 2
–3y = –2x + 2
y = 2
3
x –
Hence the slope of 2x – 3y = 2 is .
2
3
Hence the equation for our line is
y = (x – (0)) + (–3)
–3
2
Since perpendicular lines have slopes that are the
negative reciprocals of each other, our slope is .–3
2
73. d. The line that has y-intercept at –3 and is perpendicular to
the line 2x – 3y = 2.
2
3
Equations of Lines
For the slope, solve 2x – 3y = 2
–3y = –2x + 2
y = 2
3
x –
Hence the slope of 2x – 3y = 2 is .
2
3
Hence the equation for our line is
y = (x – (0)) + (–3)
–3
2
y = x – 3
–3
2
Since perpendicular lines have slopes that are the
negative reciprocals of each other, our slope is .–3
2
74. Linear Equations and Lines
Many real world relations between two quantities are linear.
For example the cost $y is a linear formula of x–the number
of apples bought.
75. Linear Equations and Lines
Many real world relations between two quantities are linear.
For example the cost $y is a linear formula of x–the number
of apples bought. For those relations that we don’t know
whether they are linear or not, linear formulas give us the
most basic “educated guesses”.
76. Linear Equations and Lines
Many real world relations between two quantities are linear.
For example the cost $y is a linear formula of x–the number
of apples bought. For those relations that we don’t know
whether they are linear or not, linear formulas give us the
most basic “educated guesses”. The following example
demonstrates that these problems are pondered by people
ancient or present alike.
77. Linear Equations and Lines
Example C. We live by a river that floods regularly. On a rock
by the river bank there is a mark indicating the highest point
the water level ever reached in the recorded time.
Many real world relations between two quantities are linear.
For example the cost $y is a linear formula of x–the number
of apples bought. For those relations that we don’t know
whether they are linear or not, linear formulas give us the
most basic “educated guesses”. The following example
demonstrates that these problems are pondered by people
ancient or present alike.
78. Linear Equations and Lines
Example C. We live by a river that floods regularly. On a rock
by the river bank there is a mark indicating the highest point
the water level ever reached in the recorded time. At 12 pm
on July 11, the water level is 28 inches from this mark.
At 8 am on July 12 the water is 18 inches from this mark.
Many real world relations between two quantities are linear.
For example the cost $y is a linear formula of x–the number
of apples bought. For those relations that we don’t know
whether they are linear or not, linear formulas give us the
most basic “educated guesses”. The following example
demonstrates that these problems are pondered by people
ancient or present alike.
79. Linear Equations and Lines
Example C. We live by a river that floods regularly. On a rock
by the river bank there is a mark indicating the highest point
the water level ever reached in the recorded time. At 12 pm
on July 11, the water level is 28 inches from this mark.
At 8 am on July 12 the water is 18 inches from this mark. Let
x be a measurement for time, and y be the distance from the
water level and the mark.
Many real world relations between two quantities are linear.
For example the cost $y is a linear formula of x–the number
of apples bought. For those relations that we don’t know
whether they are linear or not, linear formulas give us the
most basic “educated guesses”. The following example
demonstrates that these problems are pondered by people
ancient or present alike.
80. Linear Equations and Lines
Example C. We live by a river that floods regularly. On a rock
by the river bank there is a mark indicating the highest point
the water level ever reached in the recorded time. At 12 pm
on July 11, the water level is 28 inches from this mark.
At 8 am on July 12 the water is 18 inches from this mark. Let
x be a measurement for time, and y be the distance from the
water level and the mark. Find the linear equation between x
and y.
Many real world relations between two quantities are linear.
For example the cost $y is a linear formula of x–the number
of apples bought. For those relations that we don’t know
whether they are linear or not, linear formulas give us the
most basic “educated guesses”. The following example
demonstrates that these problems are pondered by people
ancient or present alike.
81. Linear Equations and Lines
Example C. We live by a river that floods regularly. On a rock
by the river bank there is a mark indicating the highest point
the water level ever reached in the recorded time. At 12 pm
on July 11, the water level is 28 inches from this mark.
At 8 am on July 12 the water is 18 inches from this mark. Let
x be a measurement for time, and y be the distance from the
water level and the mark. Find the linear equation between x
and y. At 4 pm July 12, the water level is 12 inches from the
mark, is the flood easing or intensifying?
Many real world relations between two quantities are linear.
For example the cost $y is a linear formula of x–the number
of apples bought. For those relations that we don’t know
whether they are linear or not, linear formulas give us the
most basic “educated guesses”. The following example
demonstrates that these problems are pondered by people
ancient or present alike.
82. Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation.
83. Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11.
84. Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
85. Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18.
86. Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18. We want the equation y = m(x – x1) + y1
of the line that contains the points (0, 28) and (20, 18).
87. The slope m = = = –1/2
Δy
Δx
28 – 18
0 – 20
Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18. We want the equation y = m(x – x1) + y1
of the line that contains the points (0, 28) and (20, 18).
88. The slope m = = = –1/2
Hence the linear equation is y = –1/2(x – 0) + 28 or that
y = – + 28
Δy
Δx
28 – 18
0 – 20
2
Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18. We want the equation y = m(x – x1) + y1
of the line that contains the points (0, 28) and (20, 18).
x
89. The slope m = = = –1/2
Hence the linear equation is y = –1/2(x – 0) + 28 or that
y = – + 28
Δy
Δx
28 – 18
0 – 20
2
Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18. We want the equation y = m(x – x1) + y1
of the line that contains the points (0, 28) and (20, 18).
x
At 4 pm July 12, x = 28.
90. The slope m = = = –1/2
Hence the linear equation is y = –1/2(x – 0) + 28 or that
y = – + 28
Δy
Δx
28 – 18
0 – 20
2
Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18. We want the equation y = m(x – x1) + y1
of the line that contains the points (0, 28) and (20, 18).
x
At 4 pm July 12, x = 28. According to the formula
y = – 28/2 + 28 = –14 + 28 = 14.
91. The slope m = = = –1/2
Hence the linear equation is y = –1/2(x – 0) + 28 or that
y = – + 28
Δy
Δx
28 – 18
0 – 20
2
Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18. We want the equation y = m(x – x1) + y1
of the line that contains the points (0, 28) and (20, 18).
x
At 4 pm July 12, x = 28. According to the formula
y = – 28/2 + 28 = –14 + 28 = 14. But our actual observation,
the water level is only 12 inches from the mark.
92. The slope m = = = –1/2
Hence the linear equation is y = –1/2(x – 0) + 28 or that
y = – + 28
Δy
Δx
28 – 18
0 – 20
2
Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18. We want the equation y = m(x – x1) + y1
of the line that contains the points (0, 28) and (20, 18).
x
At 4 pm July 12, x = 28. According to the formula
y = – 28/2 + 28 = –14 + 28 = 14. But our actual observation,
the water level is only 12 inches from the mark.
Hence the flood is intensifying.
93. The slope m = = = –1/2
Hence the linear equation is y = –1/2(x – 0) + 28 or that
y = – + 28
Δy
Δx
28 – 18
0 – 20
2
Equations of Lines
The easiest way to set the time measurement x is to set
x = 0 (hr) to the time of the first observation. Hence set x = 0
at 12 pm July 11. Therefore at 8 am of July 12, x = 20.
In particular, we are given that at x = 0, y = 28, and at
x = 20, y = 18. We want the equation y = m(x – x1) + y1
of the line that contains the points (0, 28) and (20, 18).
x
At 4 pm July 12, x = 28. According to the formula
y = – 28/2 + 28 = –14 + 28 = 14. But our actual observation,
the water level is only 12 inches from the mark.
Hence the flood is intensifying. The linear equation that we
found is also called a trend line and it is shown below.
94. Linear Equations and Lines
x = number of hours passed since 12 pm July 11
y = distance from the water level to the high mark
10
20
30
10 20 30
y
40 50
x
(0, 28) y = –x/2 + 28
(20, 18)
95. Linear Equations and Lines
x = number of hours passed since 12 pm July 11
y = distance from the water level to the high mark
10
20
30
10 20 30
y
40 50
x
(0, 28)
(20, 18)
y = –x/2 + 28
(28, 14)
The projected distance
for 4 pm July 12
96. Linear Equations and Lines
x = number of hours passed since 12 pm July 11
y = distance from the water level to the high mark
10
20
30
10 20 30
y
40 50
x
(0, 28)
(20, 18)
y = –x/2 + 28
(28, 14)
The projected distance
for 4 pm July 12
(28, 14)
The actual data
taken at 4 pm July 12
97. Linear Equations and Lines
Exercise A. For problems 1–8 select two points and estimate
the slope, and find an equation of each line.
1. 2. 3. 4.
5. 6. 7. 8.
98. Linear Equations and Lines
Exercise B. Draw each line that passes through the given two
points. Find the slope and an equation of the line. Identify the
vertical lines 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)
23. (3/4, –1/3), (1/3, 3/2)
Exercise C. Find the equations of the following lines.
24. The line that passes through (0, 1) and has slope 3.
25. The line that passes through (–2 ,1) and has slope –1/2.
26. The line that passes through (5, 2) and is parallel to y = x.
27. The line that passes through (–3, 2) and is perpendicular
to –x = 2y.
22. (–1/4, –5/6), (2/3, –3/2)
99. Linear Equations and Lines
Exercise D.
Find the equations of the following lines.
28. The line that passes through (0, 1), (1, –2)
31. It’s perpendicular to 2x – 4y = 1 and passes through (–2, 1)
29. 30.
32. It’s perpendicular to 3y = x with x–intercept at x = –3.
33. It has y–intercept at y = 3 and is parallel to 3y + 4x = 1.
34. It’s perpendicular to the y–axis with y–intercept at 4.
35. It has y–intercept at y = 3 and is parallel to the x axis.
36. It’s perpendicular to the x– axis containing the point (4, –3).
37. It is parallel to the y axis has x–intercept at x = –7.
38. It is parallel to the x axis has y–intercept at y = 7.
100. Linear Equations and Lines
The cost y of renting a tour boat consists of a base–cost plus
the number of tourists x. With 4 tourists the total cost is $65,
with 11 tourists the total is $86.
39. What is the base cost and what is the charge per tourist?
40. Find the equation of y in terms of x.
41. What is the total cost if there are 28 tourists?
The temperature y of water in a glass is rising slowly.
After 4 min. the temperature is 30 Co, and after 11 min. the
temperature is up to 65 Co. Answer 42–44 assuming the
temperature is rising linearly.
42. What is the temperature at time 0 and what is the rate of
the temperature rise?
43. Find the equation of y in terms of time.
44. How long will it take to bring the water to a boil at 100 Co?
101. Linear Equations and Lines
The cost of gas y on May 3 is $3.58 and on May 9 is $4.00.
Answer 45–47 assuming the price is rising linearly.
45. Let x be the date in May, what is the rate of increase in
price in terms of x?
46. Find the equation of the price in term of the date x in May.
47. What is the projected price on May 20?
48. In 2005, the most inexpensive tablet cost $900. In the year
2010, it was $500. Find the equation of the price p in terms of
time t. What is the projected price in the year 2014?