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The document discusses using the two intercept form to find the equation of a line given two points. It provides the two intercept form equation, where a and b are the x and y intercepts. It then works through three examples of finding the line equation using two points and the two intercept form. It lists additional practice problems and their solutions for finding line equations using two points and the two intercept form.

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Sum and product of roots

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.

Exponential Equation & Inequalities.pptx

The document provides instructions for students on exponential equations and inequalities. It defines exponential equations as equations with variable exponents and exponential inequalities as inequalities with exponents that can be solved similarly to equations. Examples of each are provided along with steps to solve them. Students are given problems to solve and informed that an assessment must be completed by midnight.

Factoring Polynomials

This document discusses various methods for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF) of terms in a polynomial.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose product is c and sum is b.
3) Factoring polynomials using grouping when the GCF is not a single term.
4) Recognizing perfect square trinomials and factoring using (a + b)^2 = a^2 + 2ab + b^2.

Nature of the roots and sum and product of the roots of a quadratic equation

Provides example and solutions about about how to find the roots of a quadratic equation and its sum and product.

equation of the line using two point form

This document discusses using the two-point form to find the equation of a line given two points. It provides the two-point form equation, examples of using the form to find the slope and y-intercept of lines, and practice problems for determining the equation of lines passing through two points. The goal is to determine the equation in slope-intercept form using the two-point form equation and substituting the x- and y-coordinates of the two points.

Combined Variation

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.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, was developed by the French mathematician René Descartes. It uses two perpendicular number lines, the x-axis and y-axis, that intersect at the origin (0,0) to locate points in a plane. Each point is identified with an ordered pair of numbers known as Cartesian coordinates that represent the distance from the origin on the x-axis and y-axis. The system divides the plane into four quadrants and allows points to be easily plotted and located.

Solving quadratics by completing the square

The document discusses solving quadratic equations by completing the square. It provides examples of perfect square trinomials and how to find the missing constant term to create them. The steps for solving a quadratic equation by completing the square are described: 1) move all terms to the left side, 2) find and add the "completing the square" term, 3) factor into a perfect square trinomial, 4) take the square root of each side, 5) solve the resulting equations. Additional examples demonstrate applying these steps to solve quadratic equations algebraically.

Sum and product of roots

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.

Exponential Equation & Inequalities.pptx

The document provides instructions for students on exponential equations and inequalities. It defines exponential equations as equations with variable exponents and exponential inequalities as inequalities with exponents that can be solved similarly to equations. Examples of each are provided along with steps to solve them. Students are given problems to solve and informed that an assessment must be completed by midnight.

Factoring Polynomials

This document discusses various methods for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF) of terms in a polynomial.
2) Factoring trinomials of the form x^2 + bx + c by finding two numbers whose product is c and sum is b.
3) Factoring polynomials using grouping when the GCF is not a single term.
4) Recognizing perfect square trinomials and factoring using (a + b)^2 = a^2 + 2ab + b^2.

Nature of the roots and sum and product of the roots of a quadratic equation

Provides example and solutions about about how to find the roots of a quadratic equation and its sum and product.

equation of the line using two point form

This document discusses using the two-point form to find the equation of a line given two points. It provides the two-point form equation, examples of using the form to find the slope and y-intercept of lines, and practice problems for determining the equation of lines passing through two points. The goal is to determine the equation in slope-intercept form using the two-point form equation and substituting the x- and y-coordinates of the two points.

Combined Variation

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.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, was developed by the French mathematician René Descartes. It uses two perpendicular number lines, the x-axis and y-axis, that intersect at the origin (0,0) to locate points in a plane. Each point is identified with an ordered pair of numbers known as Cartesian coordinates that represent the distance from the origin on the x-axis and y-axis. The system divides the plane into four quadrants and allows points to be easily plotted and located.

Solving quadratics by completing the square

The document discusses solving quadratic equations by completing the square. It provides examples of perfect square trinomials and how to find the missing constant term to create them. The steps for solving a quadratic equation by completing the square are described: 1) move all terms to the left side, 2) find and add the "completing the square" term, 3) factor into a perfect square trinomial, 4) take the square root of each side, 5) solve the resulting equations. Additional examples demonstrate applying these steps to solve quadratic equations algebraically.

Factoring Sum and Difference of Two Cubes

This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u

Special Products

MATH 101
SPECIAL PRODUCTS
Lecture by: Ms. Cherry Estabillo

Factoring the Sum and Difference of Two Cubes Worksheet

This document contains a mathematics worksheet for 8th grade algebra students. It provides instructions on factoring the sum and difference of two cubes using steps such as identifying common factors, taking cube roots, and forming trinomial expressions. The worksheet then lists 14 practice problems for students to factor expressions involving sums and differences of cubes.

Factoring Polynomials with common monomial factor

This document discusses common monomial factoring, which is writing a polynomial as a product of two polynomials where one is a monomial that factors each term. It provides examples of finding the greatest common factor (GCF) of terms in a polynomial and using it to factor the polynomial. Specifically, it factors polynomials like 4m^2 + 10m^4, 6x^4 + 9x^2y + 15x^5y, and 25b^3c^2 - 5b^2c.

Solving Equations Transformable to Quadratic Equation Including Rational Alge...

Provides examples and solutions on how to solve equations that is transformable to quadratic equation and rational algebraic equations.

Rational algebraic expressions

This document provides information about rational algebraic expressions. It defines a rational algebraic expression as a ratio of two polynomials where the denominator is not equal to zero. It gives examples of rational expressions like 7x/2y and 10x-5/(x+3). It explains that rational expressions are defined for all real numbers except those that would make the denominator equal to zero. The document also discusses how to simplify rational expressions by factoring the numerator and denominator and cancelling common factors. It provides examples of when cancellation is and is not allowed.

Grade 8 Mathematics Common Monomial Factoring

The document provides instructions for an individual activity where students must follow 16 directions within 2 minutes to complete tasks like writing their name, drawing shapes, and tapping their desk. It asks what implications the activity might have if directions are not followed properly. The second part of the document provides examples of factoring polynomials using greatest common factor and common monomial factoring methods. It includes practice problems for students to determine the greatest common factor, quotient, and factored form. The document emphasizes following directions and learning different factoring techniques.

Extracting the roots

The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.

Synthetic division

This document provides instructions for using synthetic division to divide polynomials. It contains the following key points:
1. Synthetic division can be used to divide polynomials when the divisor has a leading coefficient of 1 and there is a coefficient for every power of the variable in the numerator.
2. The procedure involves writing the terms of the numerator in descending order, bringing down the constant of the divisor, multiplying and adding down the columns to obtain the coefficients of the quotient polynomial and the remainder.
3. An example problem walks through each step of synthetic division to divide (5x^4 - 4x^2 + x + 6) / (x - 3), obtaining a quotient of 5x^3 + 15

Solving Quadratic Equations by Factoring

This document provides examples for solving quadratic equations by factoring. It explains how to solve equations of the form ax^2 + bx = 0 and ax^2 + bx + c = 0 by factoring and setting each factor equal to zero. Some example problems are worked out step-by-step, including solving 11x^2 - 13x = 8x - 3x^2 and 7x^2 + 18x = 10x^2 + 12x. The document also discusses using the fact that the roots of ax^2 + bx = 0 are x = 0 and x = -b/a to solve equations without factoring. It concludes by explaining how to use the zero product property to solve a quadratic

Solving Systems of Linear Equations in Two Variables by Graphing

This document discusses solving systems of linear equations in two variables by graphing. It begins by recalling the different types of systems and their properties. It then shows examples of writing linear equations in slope-intercept form and graphing individual equations. The main steps for solving a system by graphing are outlined: 1) write both equations in slope-intercept form, 2) graph them on the same plane, 3) find the point of intersection, and 4) check that the solution satisfies both equations. Several examples are worked through demonstrating how to graph systems, find the point of intersection, and verify the solution. The document concludes with an application problem asking students to solve systems, identify the solution location on a map, and describe the

Solving Equations Involving Radical Expressions

This document discusses how to solve radical equations by isolating the radical expression, removing the radical sign by raising both sides to the appropriate power, and solving the resulting equation. It provides examples of solving various radical equations step-by-step and checking solutions. Key steps include isolating the radical term, removing the radical, solving the resulting equation, and checking for extraneous roots.

Strategic Intervention Materials

This document provides an overview and activities on solving quadratic equations by factoring. It begins by defining quadratic equations and their standard form. Several activities are presented to practice identifying quadratic equations, rewriting them in standard form, and factoring trinomials of the form x^2 + bx + c. The final activity involves factoring quadratic equations to determine their roots. The document aims to build mastery of skills needed to solve quadratic equations using factoring techniques.

Square of a Binomial (Special Products)

The document discusses finding the square of a binomial expression by using the FOIL method. It explains that squaring a binomial results in a trinomial with the square of the first term, twice the product of the terms, and the square of the last term. Examples are provided of squaring binomial expressions with variables to demonstrate this perfect square trinomial pattern.

Multiplying polynomials

The document describes three methods for multiplying polynomials:
1) The distributive property, which involves multiplying each term of one polynomial with each term of the other.
2) FOIL (First, Outer, Inner, Last), which is a mnemonic for multiplying binomials by multiplying corresponding terms.
3) The box method, which involves drawing a box and writing one polynomial above and beside the box, then multiplying corresponding terms. Examples are provided to demonstrate each method.

7.8.-SPECIAL-PRODUCTS.ppt

This document discusses special products of binomials, including:
- (a + b)2 = a2 + 2ab + b2, known as a perfect-square trinomial
- (a - b)2 = a2 - 2ab + b2, also a perfect-square trinomial
- (a + b)(a - b) = a2 - b2, known as the difference of two squares
It provides examples of using these rules to simplify expressions involving binomials squared or multiplied together.

Rectangular coordinate system

This document defines and explains the rectangular coordinate system. It begins by introducing Rene Descartes as the "Father of Modern Mathematics" who developed the Cartesian plane. The Cartesian plane is composed of two perpendicular number lines that intersect at the origin (0,0) and divide the plane into four quadrants. Each point on the plane is identified by an ordered pair (x,y) denoting its distance from the x-axis and y-axis. The sign of the x and y values determines which quadrant a point falls into. Examples are provided to identify points and their corresponding quadrants.

Equation of a Circle in standard and general form

Writing an equation of a circle in standard or general form and finding the radius and the center of a circle given an equation.

Factoring general trinomials

This document provides instructions for factoring trinomials with leading coefficients of 1 or greater than 1. For trinomials with a leading coefficient of 1, the document explains how to list the factors of the last term, identify the factor pair that sums to the middle term, and write the factors. For trinomials with a leading coefficient greater than 1, the instructions are to find the product of the leading and last terms, identify factor pairs that sum to the middle term, rewrite the trinomial, group terms, and factor. Examples are provided to demonstrate the process.

Division Of Polynomials

Division of polynomials follows the same rules as division of real numbers. To divide a polynomial by a monomial, each term of the dividend is divided by the monomial divisor. To divide a polynomial by a polynomial, long division is used by repeatedly determining the quotient and remainder until the division is complete. The steps of long division are shown through an example of dividing a third degree polynomial by a linear polynomial divisor. The quotient and remainder are checked by multiplying the divisor and quotient together.

Rbsc class-9-maths-chapter-4

This document provides solutions to exercises from NCERT Class 9 Maths Chapter 4 on linear equations in two variables. It includes:
1) Solving linear equations representing word problems and expressing equations in the form ax + by + c = 0.
2) Finding solutions that satisfy given linear equations and determining the value of k if a given point is a solution.
3) Drawing graphs of various linear equations by plotting points that satisfy each equation.
4) Giving two equations of lines passing through a point and noting there are infinitely many such lines.

Equation Of A Line

This document discusses different methods for finding the equation of a line, including:
1) Given the slope and y-intercept
2) Using a graph
3) Given a point and the slope
4) Given two points
It provides examples of how to write the line equation using each method.

Factoring Sum and Difference of Two Cubes

This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u

Special Products

MATH 101
SPECIAL PRODUCTS
Lecture by: Ms. Cherry Estabillo

Factoring the Sum and Difference of Two Cubes Worksheet

This document contains a mathematics worksheet for 8th grade algebra students. It provides instructions on factoring the sum and difference of two cubes using steps such as identifying common factors, taking cube roots, and forming trinomial expressions. The worksheet then lists 14 practice problems for students to factor expressions involving sums and differences of cubes.

Factoring Polynomials with common monomial factor

This document discusses common monomial factoring, which is writing a polynomial as a product of two polynomials where one is a monomial that factors each term. It provides examples of finding the greatest common factor (GCF) of terms in a polynomial and using it to factor the polynomial. Specifically, it factors polynomials like 4m^2 + 10m^4, 6x^4 + 9x^2y + 15x^5y, and 25b^3c^2 - 5b^2c.

Solving Equations Transformable to Quadratic Equation Including Rational Alge...

Provides examples and solutions on how to solve equations that is transformable to quadratic equation and rational algebraic equations.

Rational algebraic expressions

This document provides information about rational algebraic expressions. It defines a rational algebraic expression as a ratio of two polynomials where the denominator is not equal to zero. It gives examples of rational expressions like 7x/2y and 10x-5/(x+3). It explains that rational expressions are defined for all real numbers except those that would make the denominator equal to zero. The document also discusses how to simplify rational expressions by factoring the numerator and denominator and cancelling common factors. It provides examples of when cancellation is and is not allowed.

Grade 8 Mathematics Common Monomial Factoring

The document provides instructions for an individual activity where students must follow 16 directions within 2 minutes to complete tasks like writing their name, drawing shapes, and tapping their desk. It asks what implications the activity might have if directions are not followed properly. The second part of the document provides examples of factoring polynomials using greatest common factor and common monomial factoring methods. It includes practice problems for students to determine the greatest common factor, quotient, and factored form. The document emphasizes following directions and learning different factoring techniques.

Extracting the roots

The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.

Synthetic division

This document provides instructions for using synthetic division to divide polynomials. It contains the following key points:
1. Synthetic division can be used to divide polynomials when the divisor has a leading coefficient of 1 and there is a coefficient for every power of the variable in the numerator.
2. The procedure involves writing the terms of the numerator in descending order, bringing down the constant of the divisor, multiplying and adding down the columns to obtain the coefficients of the quotient polynomial and the remainder.
3. An example problem walks through each step of synthetic division to divide (5x^4 - 4x^2 + x + 6) / (x - 3), obtaining a quotient of 5x^3 + 15

Solving Quadratic Equations by Factoring

This document provides examples for solving quadratic equations by factoring. It explains how to solve equations of the form ax^2 + bx = 0 and ax^2 + bx + c = 0 by factoring and setting each factor equal to zero. Some example problems are worked out step-by-step, including solving 11x^2 - 13x = 8x - 3x^2 and 7x^2 + 18x = 10x^2 + 12x. The document also discusses using the fact that the roots of ax^2 + bx = 0 are x = 0 and x = -b/a to solve equations without factoring. It concludes by explaining how to use the zero product property to solve a quadratic

Solving Systems of Linear Equations in Two Variables by Graphing

This document discusses solving systems of linear equations in two variables by graphing. It begins by recalling the different types of systems and their properties. It then shows examples of writing linear equations in slope-intercept form and graphing individual equations. The main steps for solving a system by graphing are outlined: 1) write both equations in slope-intercept form, 2) graph them on the same plane, 3) find the point of intersection, and 4) check that the solution satisfies both equations. Several examples are worked through demonstrating how to graph systems, find the point of intersection, and verify the solution. The document concludes with an application problem asking students to solve systems, identify the solution location on a map, and describe the

Solving Equations Involving Radical Expressions

This document discusses how to solve radical equations by isolating the radical expression, removing the radical sign by raising both sides to the appropriate power, and solving the resulting equation. It provides examples of solving various radical equations step-by-step and checking solutions. Key steps include isolating the radical term, removing the radical, solving the resulting equation, and checking for extraneous roots.

Strategic Intervention Materials

This document provides an overview and activities on solving quadratic equations by factoring. It begins by defining quadratic equations and their standard form. Several activities are presented to practice identifying quadratic equations, rewriting them in standard form, and factoring trinomials of the form x^2 + bx + c. The final activity involves factoring quadratic equations to determine their roots. The document aims to build mastery of skills needed to solve quadratic equations using factoring techniques.

Square of a Binomial (Special Products)

The document discusses finding the square of a binomial expression by using the FOIL method. It explains that squaring a binomial results in a trinomial with the square of the first term, twice the product of the terms, and the square of the last term. Examples are provided of squaring binomial expressions with variables to demonstrate this perfect square trinomial pattern.

Multiplying polynomials

The document describes three methods for multiplying polynomials:
1) The distributive property, which involves multiplying each term of one polynomial with each term of the other.
2) FOIL (First, Outer, Inner, Last), which is a mnemonic for multiplying binomials by multiplying corresponding terms.
3) The box method, which involves drawing a box and writing one polynomial above and beside the box, then multiplying corresponding terms. Examples are provided to demonstrate each method.

7.8.-SPECIAL-PRODUCTS.ppt

This document discusses special products of binomials, including:
- (a + b)2 = a2 + 2ab + b2, known as a perfect-square trinomial
- (a - b)2 = a2 - 2ab + b2, also a perfect-square trinomial
- (a + b)(a - b) = a2 - b2, known as the difference of two squares
It provides examples of using these rules to simplify expressions involving binomials squared or multiplied together.

Rectangular coordinate system

This document defines and explains the rectangular coordinate system. It begins by introducing Rene Descartes as the "Father of Modern Mathematics" who developed the Cartesian plane. The Cartesian plane is composed of two perpendicular number lines that intersect at the origin (0,0) and divide the plane into four quadrants. Each point on the plane is identified by an ordered pair (x,y) denoting its distance from the x-axis and y-axis. The sign of the x and y values determines which quadrant a point falls into. Examples are provided to identify points and their corresponding quadrants.

Equation of a Circle in standard and general form

Writing an equation of a circle in standard or general form and finding the radius and the center of a circle given an equation.

Factoring general trinomials

This document provides instructions for factoring trinomials with leading coefficients of 1 or greater than 1. For trinomials with a leading coefficient of 1, the document explains how to list the factors of the last term, identify the factor pair that sums to the middle term, and write the factors. For trinomials with a leading coefficient greater than 1, the instructions are to find the product of the leading and last terms, identify factor pairs that sum to the middle term, rewrite the trinomial, group terms, and factor. Examples are provided to demonstrate the process.

Division Of Polynomials

Division of polynomials follows the same rules as division of real numbers. To divide a polynomial by a monomial, each term of the dividend is divided by the monomial divisor. To divide a polynomial by a polynomial, long division is used by repeatedly determining the quotient and remainder until the division is complete. The steps of long division are shown through an example of dividing a third degree polynomial by a linear polynomial divisor. The quotient and remainder are checked by multiplying the divisor and quotient together.

Factoring Sum and Difference of Two Cubes

Factoring Sum and Difference of Two Cubes

Special Products

Special Products

Factoring the Sum and Difference of Two Cubes Worksheet

Factoring the Sum and Difference of Two Cubes Worksheet

Factoring Polynomials with common monomial factor

Factoring Polynomials with common monomial factor

Solving Equations Transformable to Quadratic Equation Including Rational Alge...

Solving Equations Transformable to Quadratic Equation Including Rational Alge...

Rational algebraic expressions

Rational algebraic expressions

Grade 8 Mathematics Common Monomial Factoring

Grade 8 Mathematics Common Monomial Factoring

Extracting the roots

Extracting the roots

Synthetic division

Synthetic division

Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Factoring

Solving Systems of Linear Equations in Two Variables by Graphing

Solving Systems of Linear Equations in Two Variables by Graphing

Solving Equations Involving Radical Expressions

Solving Equations Involving Radical Expressions

Strategic Intervention Materials

Strategic Intervention Materials

Square of a Binomial (Special Products)

Square of a Binomial (Special Products)

Multiplying polynomials

Multiplying polynomials

7.8.-SPECIAL-PRODUCTS.ppt

7.8.-SPECIAL-PRODUCTS.ppt

Rectangular coordinate system

Rectangular coordinate system

Equation of a Circle in standard and general form

Equation of a Circle in standard and general form

Factoring general trinomials

Factoring general trinomials

Division Of Polynomials

Division Of Polynomials

Rbsc class-9-maths-chapter-4

This document provides solutions to exercises from NCERT Class 9 Maths Chapter 4 on linear equations in two variables. It includes:
1) Solving linear equations representing word problems and expressing equations in the form ax + by + c = 0.
2) Finding solutions that satisfy given linear equations and determining the value of k if a given point is a solution.
3) Drawing graphs of various linear equations by plotting points that satisfy each equation.
4) Giving two equations of lines passing through a point and noting there are infinitely many such lines.

Equation Of A Line

This document discusses different methods for finding the equation of a line, including:
1) Given the slope and y-intercept
2) Using a graph
3) Given a point and the slope
4) Given two points
It provides examples of how to write the line equation using each method.

Linear equation in 2 variable class 10

The document discusses linear pairs of equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0. It explains that a pair of linear equations can be solved either algebraically or graphically. The graphical method involves plotting the lines defined by each equation on a graph and analyzing their intersection. Parallel lines mean no solution, intersecting lines mean a unique solution, and coincident lines mean infinitely many solutions. Several examples are worked through to demonstrate these concepts.

1554 linear equations in two variables

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

2 3 Bzca5e

The document discusses linear functions and slopes. It provides examples of finding the slope of a line between two points, writing the equation of a line in point-slope and slope-intercept form, graphing linear equations, finding the x- and y-intercepts of a line, and applications of linear functions including using a graphing calculator.

identities1.2

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.

point slope form

This document discusses using the point-slope form to find the equation of a line given a slope and point. It provides the point-slope form equation, and examples of finding the line equation for different slopes and points. Exercises are provided for the reader to practice finding additional line equations using given slopes and points.

Linear equations Class 10 by aryan kathuria

This document discusses linear equations and methods to solve systems of linear equations. It defines a linear equation as an equation 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. Systems of linear equations can have unique solutions, infinite solutions, or no solutions depending on whether the lines intersect, are coincident, or are parallel. The document describes graphical and algebraic methods to solve systems, including elimination, substitution, and cross-multiplication methods. It provides examples of using each algebraic method to solve systems of two linear equations with two unknowns.

Linear equation in two variables

This document discusses solving systems of linear equations in two variables. There are three main algebraic methods discussed: 1) elimination by substitution, which involves substituting one variable's expression into the other equation to get an equation with just one variable; 2) elimination by equating coefficients, which involves multiplying equations by constants and subtracting to eliminate one variable; and 3) cross-multiplication, which uses cross-multiplication of fractions to eliminate one variable. An example of using elimination by substitution to solve the system x + 2y = -1 and 2x - 3y = 12 is shown.

10TH MATH.pptx

This document discusses solving systems of linear equations using the cross-multiplication method. It begins by explaining the method using an example of finding the cost of oranges and apples. It then outlines the general steps of the method: (1) multiply one equation by the coefficient of the other and vice versa, (2) subtract the multiplied equations to isolate one variable, (3) substitute back to find the other variable. The document notes there are three possible outcomes: unique solution if coefficients ratios are unequal, infinitely many solutions if ratios are equal, and no solution if ratios are equal but constants are not. It provides a diagram and summary of the method and conditions for the different outcomes.

chapter1_part2.pdf

1. This document discusses linear equations, slope, graphing lines, writing equations in slope-intercept form, and solving systems of linear equations.
2. Key concepts explained include slope as rise over run, the different forms of writing a linear equation, finding the x- and y-intercepts, and using two points to write the equation of a line in slope-intercept form.
3. Examples are provided to demonstrate how to graph lines based on their equations in different forms, find intercepts, write equations from two points, and solve systems of linear equations.

Topic 11 straight line graphs

The document discusses drawing and finding equations of straight lines from tables of values and graphs. It provides examples of:
- Drawing lines from equations by making tables of x- and y-values and plotting points
- Finding the gradient and y-intercept from linear equations
- Using graphs to solve simultaneous equations by finding the point of intersection
- Finding equations of lines from their graphs by calculating gradient and y-intercept
- Drawing lines that pass through given points and are parallel to other lines.

Persamaan fungsi linier

This document discusses linear functions and systems of linear equations. It begins by defining the standard form of a linear function as y = mx + b, where m is the slope and b is the y-intercept. It then discusses various ways to determine the equation of a linear function given different inputs like slope and a point, or two points. The document also discusses how to graph linear functions and systems of linear equations. It describes three possible solutions for a system of two equations with two unknowns: a unique solution, no solution, or infinitely many solutions. Finally, it covers two methods for solving systems of linear equations: elimination and substitution.

Linear equation in 2 variables

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AN EQUATION WHICH CAN BE WRITTEN IN THE FORM OF ax+by+c=0 WHERE a,b and c ARE REAL NUMBERS.
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- 1. EQUATION OF THE LINE USINGTWO INTERCEPT FORM
- 2. Jessebel G. Bautista Antonio J. Villegas Voc’l High School My Profile
- 3. OBJECTIVES: 1.determine the values of the variables 2.use two intercept form 3.substitute the given values 4.find the equation of a line
- 4. Mathematical Concepts: Two Intercept Form where a and b are x and y intercepts of the line respectively. 𝑥 𝑎 + 𝑦 𝑏 = 1
- 5. Mathematical Concepts: The Point-Slope Form given a slope and a point of a line, we may find the equation by substituting their respective values in the point-slope form. y - y1 = m (x - x1)
- 6. Mathematical Concepts: The Two-Point Form given two points of a line determine the values of x1, y1, x2, and y2 then substitute it to the two-point form to find the equation of the line. y - y1 = 𝒚 𝟐−𝒚 𝟏 𝒙 𝟐−𝒙 𝟏 (x - x1)
- 7. EXAMPLE 1 1. (6, 0) and (0, -2) Solution: Let a = 6 b = -2 Use two intercept form since x and y intecepts are given. Find the equation of the line that passes through the following points. 𝑥 6 + 𝑦 −2 = 1 { 𝑥 6 + 𝑦 −2 = 1 } 6 x - 3y = 6 or -3y = -x + 6 y = 1 3 x + 2 𝑥 𝑎 + 𝑦 𝑏 = 1
- 8. EXAMPLE 2 2. (4, 0) and (0, 4) Solution: Let a = 4 b = 4 Use two intercept form since x and y intecepts are given. Find the equation of the line that passes through the following points. 𝑥 4 + 𝑦 4 = 1 { 𝑥 4 + 𝑦 4 = 1 } 4 x + y = 4 or y = -x + 4 𝑥 𝑎 + 𝑦 𝑏 = 1
- 9. EXAMPLE 3 3. (4,0) and (0, 2) Solution: a = 4 b = 2 Use two intercept form since x and y intecepts are given. Find the equation of the line that passes through the following points. 𝑥 4 + 𝑦 2 = 1 { 𝑥 4 + 𝑦 2 = 1 } 4 x + 2y = 4 𝑥 𝑎 + 𝑦 𝑏 = 1
- 10. TO DO … 1. (9,0) & (0, -5) 2. (3,0) & (0, 2) 3. (4,0) & (0,1) 4. (-7, 0) & (0, -3) 5. ( 4, 0) & (0, -2) 6. (8, 0) & (0, -3) 7. ( -5, 0) & (0 , 4) 8. (11, 0) & (0, -9) 9. (-6, 0) & (0, -2) 10. (4, 0) & (0, -6) Find the equation of the line that passes through the following points.
- 11. SALAMAT PO!!!
- 12. Learning Resources: Grade 8 Math Time k-to-12-grade-8-math-learner-module Learning Resources: