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René Descartes


Mathematics linear equation in two variables


Introduction What is a linear equation in two variable? The four quads of a graph. Types of pair of linear equation. Definitions. Algebraic methods of solving a pair of linear equation.

What is a linear equation?:

What is a linear equation? A "system" of equations is a set or collection of equations that you deal with all together at once. Linear equations (ones that graph as straight lines) are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables.

The four quads of a graph.:

The four quads of a graph.

Types of pair of linear equation:

Types of pair of linear equation Inconsistent pair. Consistent pair. Dependent pair.


DEFINITIONS Inconsistent pair: a pair of linear equation which has no solution, is called an inconsistent pair of linear equation.

Consistent pair:

Consistent pair A pair of linear equations in two variable which has a solution is called a consistent pair of linear equation.

Dependent pair:

Dependent pair A pair of linear equation which has infinitely many solutions is called a dependent pair of linear equation.

Algebraic Methods of Solving a Pair of Linear Equations :

Algebraic Methods of Solving a Pair of Linear Equations To find the solution to pair of linear equations , graphical method may not always give the most accurate solutions. Especially, when the point representing the solution has non-integral coordinates like ( ) or .

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There are three algebraic methods that can be used to solve a pair of linear equations namely (1) Substitution method (2) Elimination method (3) Cross - multiplication method

Substitution method: :

Substitution method: The first step to solve a pair of linear equations by the substitution method is to solve one equation for either of the variables. The choice of equation or variable in a given pair does not affect the solution for the pair of equations.

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In the next step, we’ll substitute the resultant value of one variable obtained in the other equation and solve for the other variable. In the last step, we can substitute the value obtained of the variable in any one equation to find the value of the second variable.

Elimination method: :

Elimination method : Step 1: Multiply the equations with suitable non-zero constants, so that the coefficients of one variable in both equations become equal. Step 2: Subtract one equation from another, to eliminate the variable with equal coefficients.

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Step 3: Solve for the remaining variable. Step 4: Substitute the obtained value of the variable in one of the equations and solve for the second variable

Cross multiplication Method :

Cross multiplication Method In this section we will discuss cross multiplication method. It is also known as Cramer's rule. Let the two equations be , a1x + b1y +c1 = 0 a2x + b2y +c2 = 0 be a system of linear equations in two variables x and y such that

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X Y 1 b 1 c 2 -b 2 c 1 c 1 a 2 -c 2 a 1 a 1 b 2 -a 2 b 1

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a1             b1 ------≠------ a2            b2 i.e. a1 b2 - a2 b1 ≠0 Then the system has a unique solution given by (b1c2 - b2c1)                    (c1a2 - c2a1) x = -------------    and     y = --------------- (a1b2 - a2b1)                    (a1b2 - a2b1)

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Submitted by: Tarandeep Kaur Hera Azhar Priya Bhalla Shivangi Rajput Rupanshi Shekhawat. Submitted to: Miss Manika Rastogi

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