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Solving Quadratic Equations by Completing the Square

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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

Quadratic Equations (Quadratic Formula) Using PowerPoint

This document summarizes the steps to solve a quadratic equation using the quadratic formula. It works through solving the specific equation 5y^2 - 8y + 3 = 0 as an example. The key steps are: 1) Identifying the coefficients a, b, and c; 2) Plugging these into the quadratic formula; 3) Simplifying the terms; 4) Isolating the variable to find the solutions of 1 and 0.6. These solutions are then checked by substituting them back into the original equation.

Section 14.2 multiplying and dividing rational expressions

This document contains slides about rational expressions and their applications. It discusses multiplying, dividing, and finding reciprocals of rational expressions. Examples are provided to illustrate how to multiply and divide rational expressions by multiplying or dividing the numerators and denominators. The steps for multiplying or dividing rational expressions are outlined as: 1) note the operation, 2) multiply/divide numerators and denominators, 3) factor completely, and 4) write in lowest terms.

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

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.

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.

quadratic equation

The document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the form ax2 + bx + c, where a ≠ 0. The constants a, b, and c are the quadratic, linear, and constant coefficients.
2) There are three main methods to solve quadratic equations: factoring, completing the square, or using the quadratic formula.
3) The discriminant, b2 - 4ac, determines the nature of the roots - two real roots if positive, one real root if zero, or two complex roots if negative.

Solving Quadratic Equations by Completing the Square

For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u

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

Quadratic Equations (Quadratic Formula) Using PowerPoint

This document summarizes the steps to solve a quadratic equation using the quadratic formula. It works through solving the specific equation 5y^2 - 8y + 3 = 0 as an example. The key steps are: 1) Identifying the coefficients a, b, and c; 2) Plugging these into the quadratic formula; 3) Simplifying the terms; 4) Isolating the variable to find the solutions of 1 and 0.6. These solutions are then checked by substituting them back into the original equation.

Section 14.2 multiplying and dividing rational expressions

This document contains slides about rational expressions and their applications. It discusses multiplying, dividing, and finding reciprocals of rational expressions. Examples are provided to illustrate how to multiply and divide rational expressions by multiplying or dividing the numerators and denominators. The steps for multiplying or dividing rational expressions are outlined as: 1) note the operation, 2) multiply/divide numerators and denominators, 3) factor completely, and 4) write in lowest terms.

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

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.

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.

quadratic equation

The document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the form ax2 + bx + c, where a ≠ 0. The constants a, b, and c are the quadratic, linear, and constant coefficients.
2) There are three main methods to solve quadratic equations: factoring, completing the square, or using the quadratic formula.
3) The discriminant, b2 - 4ac, determines the nature of the roots - two real roots if positive, one real root if zero, or two complex roots if negative.

linear equation system with 2 and 3 variables

The document discusses linear equation systems with 2 and 3 variables. It provides examples of linear equations with 1, 2, and 3 variables. There are 4 methods to solve a system of 2 linear equations: substitution, elimination, elimination-substitution, and graphing. Graphing involves drawing the lines represented by each equation and finding their point of intersection. For 3 variables, one variable is eliminated to obtain 2 equations, which are then solved to find the values for the remaining 2 variables. These values are then substituted back into one of the original equations to solve for the eliminated variable.

MATHS - Linear equation in two variable (Class - X) Maharashtra Board

MATHS - Linear equation in two variable
(Class - X)
Maharashtra Board
Equations/Expressions
Word Problem

Elizabeth& Valarie - Linear Function

This document defines and provides examples of linear functions. It begins by defining a linear function as one that can be written in the form F(x)=ax+b, where a and b are real numbers. It notes that when written as Ax+By=C, it is in standard form. The graph of a linear function is a straight line. Examples are then provided of graphing the linear function 5x-2y=10 by finding its x- and y-intercepts and using those points to draw the line. Additional examples demonstrate finding multiple points on the line to check the solution. Pictures show how the roof of a bridge can be modeled by the linear function y=0.25x+2. A

Quadratic Equation

A quick intro. to Quadratic Equation.
Few examples have also been discussed to make the student grasp the concept easily.

QUADRATIC EQUATIONS

This document discusses four methods for solving quadratic equations: factorization, completing the square, using a formula, and using graphs. It provides an example of solving the equation 2x^2 - 10x + 12 + x^2 + 6x = -9 by factorizing into (x - 3)(x - 1) = 0, finding that the solutions are x = 3 or x = 1.

Quadratic Equation and discriminant

This document introduces the quadratic formula as a method for solving quadratic equations. It shows the steps for deriving the formula from completing the square and provides examples of its use. The discriminant is defined as b^2 - 4ac from the quadratic formula. The sign of the discriminant determines the number and type of roots: positive discriminant yields two real roots, zero discriminant yields one real root, and negative discriminant yields two complex roots. Examples are provided to illustrate each case.

Factoring polynomials

Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.

Linear Equations in Two Variables

The document discusses linear equations in two variables. It defines a linear equation as one that can be written in the standard form Ax + By = C, where A, B, and C are real numbers and A and B cannot both be zero. Examples are provided of determining if equations are linear and identifying the A, B, and C components if they are linear. The document also discusses using ordered pairs as solutions to linear equations and finding multiple solutions to a given linear equation.

Linear inequalities in two variables

The document discusses linear inequalities in two variables and their graphical representations. It introduces the Cartesian coordinate system developed by Rene Descartes and its importance. It explains how to graph linear inequalities by first drawing the line as an equation, then determining whether to shade above or below the line based on whether a test point satisfies the inequality. Students are assigned to bring graphing paper, coloring materials, and a ruler to class on Monday to graph linear inequalities.

Quadratic equation slideshare

This document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the general form of ax2 + bx + c = 0, where a ≠ 0.
2) It discusses the importance of quadratic equations, noting that the term "quadratic" comes from the variable being squared (x2) and that a quadratic equation is a trinomial expression with three terms.
3) It presents the method of factorization to solve quadratic equations, showing that if ax2 + bx + c = (rx + p)(sx + q) = 0, then the solutions are x1 = -p/r and x2 = -q/s

Linear equation in two variable

The document discusses methods for solving systems of linear equations in two variables:
1) Graphical method involves plotting the lines defined by each equation on a graph and finding their point of intersection.
2) Algebraic methods include substitution, elimination by equating coefficients, and cross-multiplication. Elimination involves manipulating the equations to eliminate one variable and solve for the other.
3) Examples demonstrate solving a system using substitution and elimination to find the solution values for x and y.

16.2 Solving by Factoring

This document provides an overview of solving quadratic equations by factoring. It begins with the standard form of a quadratic equation and explains the zero factor property. Examples are provided to demonstrate factoring quadratic equations and setting each factor equal to zero to solve. The steps for solving a quadratic equation by factoring are outlined. Additional examples demonstrate solving real world application problems involving quadratic equations.

Linear equations in two variables- By- Pragyan

This is a power point presentation on linear equations in two variables for class 10th. I have spent 3 hours on making this and all the equations you will see are written by me.

solving quadratic equations using quadratic formula

The document discusses how to use the quadratic formula to solve quadratic equations. It provides the formula: x = (-b ± √(b2 - 4ac)) / 2a. It then works through examples of writing quadratic equations in standard form (ax2 + bx + c = 0) and using the formula to solve them. Specifically, it solves the equations: 1) 1x2 + 3x - 27 = 0, 2) 2x2 + 7x + 5 = 0, 3) x2 - 2x = 8, and 4) x2 - 7x = 10. It concludes by providing 3 additional equations to solve using the quadratic formula.

Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic Equations

This powerpoint presentation discusses or talks about the topic or lesson Roots and Coefficients of Quadratic Equations. It also discusses and explains the rules, steps and examples of Roots and Coefficients of Quadratic Equations

Notes solving polynomial equations

The document provides information about solving polynomial equations. There are three main ways to solve polynomial equations: 1) Using factoring and the zero product property, 2) Using a graphing calculator to graph the equation, and 3) Using synthetic division. The maximum number of solutions a polynomial equation can have is equal to the degree of the polynomial. Examples are provided to demonstrate solving polynomial equations by factoring.

writing linear equation

This document provides examples for rewriting linear equations between the slope-intercept form (y=mx+b) and standard form (Ax + By = C).
It begins with examples of rewriting equations from standard form to slope-intercept form and identifying the slope (m) and y-intercept (b). Then it provides examples of rewriting from slope-intercept form to standard form. Finally, it provides a series of practice problems for rewriting linear equations between the two forms.

Adding and subtracting rational expressions

Using rules for fractions, rational expressions can be added and subtracted by finding common denominators. To find the common denominator, we find the least common multiple (LCM) of the denominators. With polynomials, the LCM will contain all factors of each denominator. We can then convert the fractions to equivalent forms using the LCM as the new denominator before combining like terms to evaluate the expression. Special cases may involve fractions with understood denominators of 1 or similar but non-equal denominators that can be made equal through factoring.

Polynomial Function and Synthetic Division

This presentation explains the basic information about Polynomial Function and Synthetic Division. Examples were given about easy ways to divide polynomial function using synthetic division. It also contains the steps on how to perform the division method of polynomial functions.

quadratic equations-1

Okay, let's solve this step-by-step:
1) Given: x = 2 and x = 3 are roots of the equation 3x^2 - 2mx + 2n = 0
2) Substitute x = 2 in the equation: 3(2)^2 - 2m(2) + 2n = 0
=> 12 - 4m + 2n = 0
3) Substitute x = 3 in the equation: 3(3)^2 - 2m(3) + 2n = 0
=> 27 - 6m + 2n = 0
4) Solve the two equations simultaneously to find m and n:
12 - 4m + 2n = 0
27 - 6m +

Linear equations in two variables

This presentation include various methods of solving linear equations like substitution, elimination and cross-multiplication method.

CLASS 9 LINEAR EQUATIONS IN TWO VARIABLES PPT

This document provides information about linear equations in two variables. It defines linear equations and explains that a linear equation in two variables can be written in the form ax + by = c. The document also discusses finding the solutions of linear equations, graphing linear equations, and equations of lines parallel to the x-axis and y-axis. Examples are provided to illustrate key concepts. In the summary, key points are restated such as linear equations having infinitely many solutions and the graph of a linear equation being a straight line.

linear equation system with 2 and 3 variables

The document discusses linear equation systems with 2 and 3 variables. It provides examples of linear equations with 1, 2, and 3 variables. There are 4 methods to solve a system of 2 linear equations: substitution, elimination, elimination-substitution, and graphing. Graphing involves drawing the lines represented by each equation and finding their point of intersection. For 3 variables, one variable is eliminated to obtain 2 equations, which are then solved to find the values for the remaining 2 variables. These values are then substituted back into one of the original equations to solve for the eliminated variable.

MATHS - Linear equation in two variable (Class - X) Maharashtra Board

MATHS - Linear equation in two variable
(Class - X)
Maharashtra Board
Equations/Expressions
Word Problem

Elizabeth& Valarie - Linear Function

This document defines and provides examples of linear functions. It begins by defining a linear function as one that can be written in the form F(x)=ax+b, where a and b are real numbers. It notes that when written as Ax+By=C, it is in standard form. The graph of a linear function is a straight line. Examples are then provided of graphing the linear function 5x-2y=10 by finding its x- and y-intercepts and using those points to draw the line. Additional examples demonstrate finding multiple points on the line to check the solution. Pictures show how the roof of a bridge can be modeled by the linear function y=0.25x+2. A

Quadratic Equation

A quick intro. to Quadratic Equation.
Few examples have also been discussed to make the student grasp the concept easily.

QUADRATIC EQUATIONS

This document discusses four methods for solving quadratic equations: factorization, completing the square, using a formula, and using graphs. It provides an example of solving the equation 2x^2 - 10x + 12 + x^2 + 6x = -9 by factorizing into (x - 3)(x - 1) = 0, finding that the solutions are x = 3 or x = 1.

Quadratic Equation and discriminant

This document introduces the quadratic formula as a method for solving quadratic equations. It shows the steps for deriving the formula from completing the square and provides examples of its use. The discriminant is defined as b^2 - 4ac from the quadratic formula. The sign of the discriminant determines the number and type of roots: positive discriminant yields two real roots, zero discriminant yields one real root, and negative discriminant yields two complex roots. Examples are provided to illustrate each case.

Factoring polynomials

Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.

Linear Equations in Two Variables

The document discusses linear equations in two variables. It defines a linear equation as one that can be written in the standard form Ax + By = C, where A, B, and C are real numbers and A and B cannot both be zero. Examples are provided of determining if equations are linear and identifying the A, B, and C components if they are linear. The document also discusses using ordered pairs as solutions to linear equations and finding multiple solutions to a given linear equation.

Linear inequalities in two variables

The document discusses linear inequalities in two variables and their graphical representations. It introduces the Cartesian coordinate system developed by Rene Descartes and its importance. It explains how to graph linear inequalities by first drawing the line as an equation, then determining whether to shade above or below the line based on whether a test point satisfies the inequality. Students are assigned to bring graphing paper, coloring materials, and a ruler to class on Monday to graph linear inequalities.

Quadratic equation slideshare

This document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the general form of ax2 + bx + c = 0, where a ≠ 0.
2) It discusses the importance of quadratic equations, noting that the term "quadratic" comes from the variable being squared (x2) and that a quadratic equation is a trinomial expression with three terms.
3) It presents the method of factorization to solve quadratic equations, showing that if ax2 + bx + c = (rx + p)(sx + q) = 0, then the solutions are x1 = -p/r and x2 = -q/s

Linear equation in two variable

The document discusses methods for solving systems of linear equations in two variables:
1) Graphical method involves plotting the lines defined by each equation on a graph and finding their point of intersection.
2) Algebraic methods include substitution, elimination by equating coefficients, and cross-multiplication. Elimination involves manipulating the equations to eliminate one variable and solve for the other.
3) Examples demonstrate solving a system using substitution and elimination to find the solution values for x and y.

16.2 Solving by Factoring

This document provides an overview of solving quadratic equations by factoring. It begins with the standard form of a quadratic equation and explains the zero factor property. Examples are provided to demonstrate factoring quadratic equations and setting each factor equal to zero to solve. The steps for solving a quadratic equation by factoring are outlined. Additional examples demonstrate solving real world application problems involving quadratic equations.

Linear equations in two variables- By- Pragyan

This is a power point presentation on linear equations in two variables for class 10th. I have spent 3 hours on making this and all the equations you will see are written by me.

solving quadratic equations using quadratic formula

The document discusses how to use the quadratic formula to solve quadratic equations. It provides the formula: x = (-b ± √(b2 - 4ac)) / 2a. It then works through examples of writing quadratic equations in standard form (ax2 + bx + c = 0) and using the formula to solve them. Specifically, it solves the equations: 1) 1x2 + 3x - 27 = 0, 2) 2x2 + 7x + 5 = 0, 3) x2 - 2x = 8, and 4) x2 - 7x = 10. It concludes by providing 3 additional equations to solve using the quadratic formula.

Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic Equations

This powerpoint presentation discusses or talks about the topic or lesson Roots and Coefficients of Quadratic Equations. It also discusses and explains the rules, steps and examples of Roots and Coefficients of Quadratic Equations

Notes solving polynomial equations

The document provides information about solving polynomial equations. There are three main ways to solve polynomial equations: 1) Using factoring and the zero product property, 2) Using a graphing calculator to graph the equation, and 3) Using synthetic division. The maximum number of solutions a polynomial equation can have is equal to the degree of the polynomial. Examples are provided to demonstrate solving polynomial equations by factoring.

writing linear equation

This document provides examples for rewriting linear equations between the slope-intercept form (y=mx+b) and standard form (Ax + By = C).
It begins with examples of rewriting equations from standard form to slope-intercept form and identifying the slope (m) and y-intercept (b). Then it provides examples of rewriting from slope-intercept form to standard form. Finally, it provides a series of practice problems for rewriting linear equations between the two forms.

Adding and subtracting rational expressions

Using rules for fractions, rational expressions can be added and subtracted by finding common denominators. To find the common denominator, we find the least common multiple (LCM) of the denominators. With polynomials, the LCM will contain all factors of each denominator. We can then convert the fractions to equivalent forms using the LCM as the new denominator before combining like terms to evaluate the expression. Special cases may involve fractions with understood denominators of 1 or similar but non-equal denominators that can be made equal through factoring.

Polynomial Function and Synthetic Division

This presentation explains the basic information about Polynomial Function and Synthetic Division. Examples were given about easy ways to divide polynomial function using synthetic division. It also contains the steps on how to perform the division method of polynomial functions.

quadratic equations-1

Okay, let's solve this step-by-step:
1) Given: x = 2 and x = 3 are roots of the equation 3x^2 - 2mx + 2n = 0
2) Substitute x = 2 in the equation: 3(2)^2 - 2m(2) + 2n = 0
=> 12 - 4m + 2n = 0
3) Substitute x = 3 in the equation: 3(3)^2 - 2m(3) + 2n = 0
=> 27 - 6m + 2n = 0
4) Solve the two equations simultaneously to find m and n:
12 - 4m + 2n = 0
27 - 6m +

linear equation system with 2 and 3 variables

linear equation system with 2 and 3 variables

MATHS - Linear equation in two variable (Class - X) Maharashtra Board

MATHS - Linear equation in two variable (Class - X) Maharashtra Board

Elizabeth& Valarie - Linear Function

Elizabeth& Valarie - Linear Function

Quadratic Equation

Quadratic Equation

QUADRATIC EQUATIONS

QUADRATIC EQUATIONS

Quadratic Equation and discriminant

Quadratic Equation and discriminant

Factoring polynomials

Factoring polynomials

Linear Equations in Two Variables

Linear Equations in Two Variables

Linear inequalities in two variables

Linear inequalities in two variables

Quadratic equation slideshare

Quadratic equation slideshare

Linear equation in two variable

Linear equation in two variable

16.2 Solving by Factoring

16.2 Solving by Factoring

Linear equations in two variables- By- Pragyan

Linear equations in two variables- By- Pragyan

solving quadratic equations using quadratic formula

solving quadratic equations using quadratic formula

Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic Equations

Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic Equations

Notes solving polynomial equations

Notes solving polynomial equations

writing linear equation

writing linear equation

Adding and subtracting rational expressions

Adding and subtracting rational expressions

Polynomial Function and Synthetic Division

Polynomial Function and Synthetic Division

quadratic equations-1

quadratic equations-1

Linear equations in two variables

This presentation include various methods of solving linear equations like substitution, elimination and cross-multiplication method.

CLASS 9 LINEAR EQUATIONS IN TWO VARIABLES PPT

This document provides information about linear equations in two variables. It defines linear equations and explains that a linear equation in two variables can be written in the form ax + by = c. The document also discusses finding the solutions of linear equations, graphing linear equations, and equations of lines parallel to the x-axis and y-axis. Examples are provided to illustrate key concepts. In the summary, key points are restated such as linear equations having infinitely many solutions and the graph of a linear equation being a straight line.

Linear Equations Ppt

The document discusses linear functions and graphs. It explains that linear graphs form straight lines and linear expressions only contain one variable with no exponents. It also defines slope as the rate of incline or decline of a line and discusses how to find the slope and y-intercept of a linear equation in slope-intercept form. Finally, it provides an example of using a linear equation to generate a table of x-y points and graph those points on a line.

linear equation

1) The document discusses linear equations in two variables, including defining their form as ax + by = c, explaining that they have infinitely many solutions, and noting that their graphs are straight lines.
2) Specific topics covered include finding solutions, drawing graphs, identifying equations for lines parallel to the x-axis and y-axis, and providing examples of writing and solving linear equations.
3) The summary restates the key points about the properties of linear equations in two variables, such as their graphical and algebraic representations.

LINEAR EQUATION IN TWO VARIABLES PPT

The document provides information about solving linear equations and 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. It discusses three methods for solving a pair of linear equations:
1) The graphical method involves plotting the equations on a graph and finding their point of intersection.
2) The algebraic methods include substitution, elimination, and cross-multiplication. Substitution involves solving one equation for one variable and substituting it into the other equation. Elimination involves eliminating one variable to obtain an equation with just one variable.
3) Cross-

2015 Upload Campaigns Calendar - SlideShare

Each month, join us as we highlight and discuss hot topics ranging from the future of higher education to wearable technology, best productivity hacks and secrets to hiring top talent. Upload your SlideShares, and share your expertise with the world!

What to Upload to SlideShare

Not sure what to share on SlideShare?
SlideShares that inform, inspire and educate attract the most views. Beyond that, ideas for what you can upload are limitless. We’ve selected a few popular examples to get your creative juices flowing.

Getting Started With SlideShare

SlideShare is a global platform for sharing presentations, infographics, videos and documents. It has over 18 million pieces of professional content uploaded by experts like Eric Schmidt and Guy Kawasaki. The document provides tips for setting up an account on SlideShare, uploading content, optimizing it for searchability, and sharing it on social media to build an audience and reputation as a subject matter expert.

Linear equations in two variables

Linear equations in two variables

CLASS 9 LINEAR EQUATIONS IN TWO VARIABLES PPT

CLASS 9 LINEAR EQUATIONS IN TWO VARIABLES PPT

Linear Equations Ppt

Linear Equations Ppt

linear equation

linear equation

LINEAR EQUATION IN TWO VARIABLES PPT

LINEAR EQUATION IN TWO VARIABLES PPT

2015 Upload Campaigns Calendar - SlideShare

2015 Upload Campaigns Calendar - SlideShare

What to Upload to SlideShare

What to Upload to SlideShare

Getting Started With SlideShare

Getting Started With SlideShare

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.

Simultaneous equations elimination 3

Simultaneous equation by elimination when no pair of variable has equal coefficient but with same sign.

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.

Maths ppt linear equations in two variables

AN EQUATION WHICH CAN BE WRITTEN IN THE FORM OF ax+by+c=0 WHERE a,b and c ARE REAL NUMBERS.
YOU WILL GET TO KNOW HOW TO REPRESENT THE EQUATIONS IN A GRAPH.

Linear equations

The document discusses various methods for solving systems of simultaneous linear equations with two variables. It explains that a system contains two or more linear equations involving the same variables. Common methods covered include substitution, where one variable is solved for and substituted into the other equation, and elimination, where coefficients are multiplied and equations are combined to eliminate one variable. Examples are provided to demonstrate both methods step-by-step. It emphasizes that solutions found must satisfy both original equations.

Chapter 2

This document provides an overview of quadratic equations and inequalities. It defines quadratic equations as equations of the form ax2 + bx + c = 0, where a, b, and c are real number constants and a ≠ 0. Examples of quadratic equations are provided. Methods for solving quadratic equations are discussed, including factoring, completing the square, and the quadratic formula. Properties of inequalities are outlined. The chapter also covers solving polynomial and rational inequalities, as well as equations and inequalities involving absolute value. Practice problems are included at the end.

Solving Linear Equations

This document summarizes three methods for solving systems of linear equations: graphing, substitution, and elimination. It provides examples of solving systems of two equations using each method. Graphing involves plotting the lines defined by each equation on a coordinate plane and finding their point of intersection. Substitution involves isolating a variable in one equation and substituting it into the other equation. Elimination involves adding or subtracting multiples of equations to remove a variable and solve for the remaining variable.

Linear equation in 2 variables

This document provides information about linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It discusses using the rectangular coordinate system to graph linear equations by plotting the x- and y-intercepts. It also describes how to determine if an ordered pair is a solution to a linear equation by substituting the x- and y-values into the equation. Finally, it briefly outlines common methods for solving systems of linear equations, including elimination, substitution, and cross-multiplication.

Bonus math project

This document provides an overview of different methods for solving quadratic equations, including factoring, graphing, using the quadratic formula, and more. It begins by defining the general and standard forms of quadratic equations. It then explains how to write equations in standard form and discusses concepts like the vertex, completing the square, determining zeroes/roots, and the discriminant. Finally, it reviews several methods that can be used to find roots of quadratic and higher-order polynomial equations, such as factoring, graphing, the quadratic formula, synthetic division, and the remainder/factor theorems.

linear equations in two variables

This document discusses linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides examples of linear equations and discusses how to graph them by plotting the x and y intercepts. It also explains how to determine if a given ordered pair is a solution to a linear equation by substituting the x and y values into the equation. Finally, it discusses different methods for solving systems of linear equations, including substitution and elimination.

linear equation in 2 variables

1) A linear equation in two variables 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.
2) The solution to a linear equation is the ordered pair that satisfies the equation when substituted into it.
3) Linear equations can be graphed on a Cartesian plane by plotting the solutions as points and connecting them.

algebra lesson notes (best).pdf

This document discusses different methods for solving simultaneous equations and quadratic equations, including:
- Solving simultaneous equations with three unknowns using matrices, determinants, or Cramer's rule. Examples are provided.
- Solving quadratic equations by factorizing (if possible), completing the square, using the quadratic formula, or graphically. Factorization is introduced as the simplest method when applicable.
- Key steps are outlined for each method, such as writing the equations in standard form and determining relevant determinants or matrices. Applications to circuit analysis and mechanical systems are mentioned.

Analytic Geometry Period 1

The document provides an overview of various topics in analytic geometry, including circle equations, distance equations, systems of two and three variable equations, linear inequalities, rational inequalities, and intersections of inequalities. It defines key concepts, provides examples of how to solve different types of problems, and notes things to remember when working with inequalities.

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.

Solving Quadratic Equations by Factoring

The document discusses solving quadratic equations by factoring. It provides examples of factoring quadratic expressions to find the solutions to the equations. These include using the zero product rule, factoring a common factor, and factoring a perfect square. It also provides two word problems involving consecutive integers and the Pythagorean theorem and shows how to set up and solve the quadratic equations derived from the word problems.

Solving linear systems by the substitution method

The document discusses solving systems of linear equations using substitution and elimination methods. It provides 4 examples of solving systems of 2 equations with 2 unknowns. The substitution method involves solving one equation for one variable in terms of the other and substituting it into the second equation. The elimination method involves multiplying equations by constants and adding/subtracting them to eliminate one variable. Both methods are shown to yield the solution point that satisfies both equations.

Linear Equations

- A linear system includes two or more equations with two or more variables. When two equations are used to model a problem, it is called a linear system.
- Common methods to solve linear systems include graphing the equations to find their intersection point, substitution where one variable is solved for in one equation and substituted into the other, and elimination where equations are combined by multiplication to eliminate a variable.
- The Hill cipher is a method to encrypt plaintext messages by performing matrix multiplication on the message represented as numbers with an encryption key matrix.

Straight-Line-Graphs-Final -2.pptx

This document contains 5 math problems involving factorizing expressions, solving equations, evaluating expressions for given values, expanding expressions, and finding the highest common factor. It also provides context on working with straight line graphs, including finding the gradient and y-intercept of a line from its equation, finding the gradient between two points, finding the midpoint and a point that divides a line segment in a given ratio, and finding the x- and y-intercepts of a line.

Maths

This document discusses linear equations in two variables. It defines linear equations in two variables as equations of the form ax + by = c, where a, b, and c are real numbers and a and b are not both zero. It explains that the graph of any linear equation in two variables is a straight line. It also categorizes different types of systems of linear equations based on the relationship between the lines: intersecting lines have a unique solution; coincident lines have an infinite number of solutions; and parallel lines have no solution. Methods for solving systems of linear equations like substitution, elimination, and graphing are also covered.

Linear, quardratic equations

This document provides an introduction to different types of equations including linear equations, simultaneous equations, and quadratic equations. It defines an equation as a statement of equality between two quantities. Linear equations are those where the highest power of the unknown is one. Simultaneous equations contain two or more unknowns and can be solved using substitution or row operations. Quadratic equations contain terms with powers of the unknown up to two and can be solved using factorization, completing the square, or the quadratic formula. Examples are provided for solving each type of equation. The objectives and end questions review solving these different equation types.

Linear equation in 2 variable class 10

Linear equation in 2 variable class 10

Simultaneous equations elimination 3

Simultaneous equations elimination 3

10TH MATH.pptx

10TH MATH.pptx

Maths ppt linear equations in two variables

Maths ppt linear equations in two variables

Linear equations

Linear equations

Chapter 2

Chapter 2

Solving Linear Equations

Solving Linear Equations

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Linear equation in 2 variables

Bonus math project

Bonus math project

linear equations in two variables

linear equations in two variables

linear equation in 2 variables

linear equation in 2 variables

algebra lesson notes (best).pdf

algebra lesson notes (best).pdf

Analytic Geometry Period 1

Analytic Geometry Period 1

Rbsc class-9-maths-chapter-4

Rbsc class-9-maths-chapter-4

Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Factoring

Solving linear systems by the substitution method

Solving linear systems by the substitution method

Linear Equations

Linear Equations

Straight-Line-Graphs-Final -2.pptx

Straight-Line-Graphs-Final -2.pptx

Maths

Maths

Linear, quardratic equations

Linear, quardratic equations

writing about opinions about Australia the movie

writing about opinions about Australia the movie

A Strategic Approach: GenAI in Education

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.

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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.

How to Add Chatter in the odoo 17 ERP Module

In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.

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Presentation required for the master in Education.

RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3

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A Survey of Techniques for Maximizing LLM Performance.pptx

A Survey of Techniques for Maximizing LLM Performance

June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...

Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202

Film vocab for eal 3 students: Australia the movie

film vocab esl

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Natural birth techniques - Mrs.Akanksha Trivedi Rama University

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This is part 2 of my Java Learning Journey. This contains Hashing, ArrayList, LinkedList, Date and Time Classes, Calendar Class and more.

PCOS corelations and management through Ayurveda.

This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.

Introduction to AI for Nonprofits with Tapp Network

Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.

Main Java[All of the Base Concepts}.docx

This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.

বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf

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 ...

DRUGS AND ITS classification slide share

Any substance (other than food) that is used to prevent, diagnose, treat, or relieve symptoms of a
disease or abnormal condition

Aficamten in HCM (SEQUOIA HCM TRIAL 2024)

SEQUOIA HCM TRIAL 2024 NEJM

The History of Stoke Newington Street Names

Presented at the Stoke Newington Literary Festival on 9th June 2024
www.StokeNewingtonHistory.com

Lapbook sobre os Regimes Totalitários.pdf

Lapbook sobre o Totalitarismo.

writing about opinions about Australia the movie

writing about opinions about Australia the movie

A Strategic Approach: GenAI in Education

A Strategic Approach: GenAI in Education

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The Diamonds of 2023-2024 in the IGRA collection

How to Add Chatter in the odoo 17 ERP Module

How to Add Chatter in the odoo 17 ERP Module

Thesis Statement for students diagnonsed withADHD.ppt

Thesis Statement for students diagnonsed withADHD.ppt

RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3

RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3

A Survey of Techniques for Maximizing LLM Performance.pptx

A Survey of Techniques for Maximizing LLM Performance.pptx

June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...

June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...

Film vocab for eal 3 students: Australia the movie

Film vocab for eal 3 students: Australia the movie

ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf

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Natural birth techniques - Mrs.Akanksha Trivedi Rama University

Natural birth techniques - Mrs.Akanksha Trivedi Rama University

Advanced Java[Extra Concepts, Not Difficult].docx

Advanced Java[Extra Concepts, Not Difficult].docx

PCOS corelations and management through Ayurveda.

PCOS corelations and management through Ayurveda.

Introduction to AI for Nonprofits with Tapp Network

Introduction to AI for Nonprofits with Tapp Network

Main Java[All of the Base Concepts}.docx

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বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf

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Lapbook sobre os Regimes Totalitários.pdf

- 3. A pair of linear equations in two variables is said to form a system of linear equations. For Example, 2x-3y+4=0 x+7y+1=0 Form a system of two linear equations in variables x and y.
- 4. The general form of linear equations in two variables x and y is ax+by+c=0, where a=/=0, b=/=0 and a,b,c are real numbers.
- 5. Pair of lines Comparison of ratios Graphical Represen- tation Algebraic Interpre- tation a1 a2 b1 b2 c2 c1 x-2y=0 1 -2 0 =/= Intersect- Unique 3x+4y-20=0 3 4 -20 lines solution x+2y-4=0 1 2 -4 = =/= Parallel No solution 2x+4y-12=0 2 4 -12 lines 2x+3y-9=0 2 3 -9 = = Coincident Infinitely 4x+6y-18=0 4 6 -18 lines many solutions a1 a2 b1 b2 a1 b1 b1 c1 a1 c1 c2a2 a2 b2 b2 c2
- 6. From the table, we can observe that if the lines represented by the equation - a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 are Intersecting lines, then = b1 a2 b2 a1
- 7. are Parallel lines, then = =/= Coincident Lines, then = = b1 c1 a1 c1b1 b2 c2b2 b2 a2 c2 a1
- 9. Lines of any given equation may be of three types - Intersecting Lines Parallel Lines Coincident Lines
- 11. Let us consider the following system of linear equations in two variable 2x-y=-1 and 3x+2y=9 Here we assign any value to one of the two variables and then determine the value of the other variable from the given equation.
- 12. For the equation 2x-y=-1 ………(1) 2x+1=y y =2x+1 3x+2y=9 ………(2) 2y=9-3x 9-3x y= ------- 2 x 0 2 y 1 5 x 3 -1 y 0 6
- 15. Let us consider the following system of linear equations in two variable x+2y=4 and 2x+4y=12 Here we assign any value to one of the two variables and then determine the value of the other variable from the given equation.
- 16. For the equation x+2y=4 ………(1) 2y=4-x y = 4-x 2 2x+4y=12 ………(2) 2x=12-4y x = 12-4y 2 x 0 4 y 2 0 x 0 6 y 3 0
- 19. Let us consider the following system of linear equations in two variable 2x+3y=9 and 4x+6y=18 Here we assign any value to one of the two variables and then determine the value of the other variable from the given equation.
- 20. For the equation 2x+3y=9 ………(1) 3y=9-2x y = 9-2x 3 4x+6y=18 ………(2) 6y=18-4x y = 18-4x 6 x 0 4.5 y 3 0 x 0 3 y 3 1
- 22. A pair of linear equation in two variables, which has a unique solution, is called a consistent pair of linear equation. A pair of linear equation in two variables, which has no solution, is called a inconsistent pair of linear equation. A pair of linear equation in two variables, which has infinitely many solutions, is called a consistent or dependent pair of linear equation.
- 24. There are three algebraic methods for solving a pair of equations :- Substitution method Elimination method Cross-multiplication method
- 26. Let the equations be :- a1x + b1y + c1 = 0 ………. (1) a2x + b2y + c2 = 0 ……….. (2) Choose either of the two equations say (1) and find the value of one variable, say y in terms of x. Now, substitute the value of y obtained in the previous step in equation (2) to get an equation in x.
- 27. Solve the equations obtained in the previous step to get the value of x. Then, substitute the value of x and get the value of y. Let us take an example :- x+2y=-1 ………(1) 2x-3y=12………(2) By eq. (1) x+2y=-1 x= -2y-1……(3)
- 28. Substituting the value of x in eq.(2), we get 2x-3y=12 2(-2y-1)-3y=12 -4y-2-3y=12 -7y=14 Y=-2 Putting the value of y in eq.(3), we get x=-2y-1 x=-2(-2)-1 x=4-1 x=3 Hence, the solution of the equation is (3,-2).
- 30. In this method, we eliminate one of the two variables to obtain an equation in one variable which can be easily solved. Putting the value of this variable in any of the given equations, the value of the other variable can be obtained. Let us take an example :- 3x+2y=11……….(1) 2x+3y=4………(2)
- 31. Multiply 3 in eq.(1) and 2 in eq.(2) and by subtracting eq.(4) from (3), we get 9x+6y=33…………(3) 4x+6y=08………(4) 5x=25 ⇒x=5 Putting the value of x in eq.(2), we get
- 32. 2x+3y=4 2(5)+3y=4 10+3y=4 3y=4-10 3y=-6 y= -2 Hence, the solution of the equation is (5,-2)
- 34. Let the equations be :- a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 Then, x = y = 1 b1 c2 - b2 c1 c1a2 - c2 a1 a1b2 - a2b1
- 35. Or :- a1x + b1y = c1 a2x + b2y = c2 Then, x = y = -1 b1 c2 - b2 c1 c1a2 - c2 a1 a1b2 - a2b1
- 36. In this method, we have put the values of a1,a2,b1,b2,c1 and c2 and by solving it, we will get the value of x and y. Let us take an example :- 2x+3y=46 3x+5y=74 i.e. 2x+3y-46=0 ………(1) 3x+5y-74=0………(2)
- 37. Then, x = y = 1 x = y = 1 (3)(-74)–(5)(-46) (-46)(3)-(-74)(2) (2)(5)-(3)(3) x = y = 1 (-222+230) (-138+148) (10-9) b1 c2 - b2 c1 c1a2 - c2 a1 a1b2 - a2b1
- 38. so, x = y = 1 8 10 1 x = 1 and y = 1 8 1 10 1 x=8 and y=10 So, Solution of the equation is (8,10)