# Algebraic Methods of Solving a Pair of Linear Equations

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1. Substitution method

If we have a pair of Linear Equations with two variables x and y, then we have to follow these steps to solve them with the substitution method-

Step 1: We have to choose any one equation and find the value of one variable in terms of other variable i.e. y in terms of x.

Step 2: Then substitute the calculated value of y in terms of x in the other equation.

Step 3: Now solve this Linear Equation in terms of x as it is in one variable only i.e. x.

Step 4: Substitute the calculate value of x in the given equations and find the value of y.

EXAMPLE

y – 2x = 1

x + 2y = 12

y = 2x + 1.

Substitute y in the second equation to get x.

x + 2 × (2x + 1) = 12

⇒ 5x + 2 = 12

x + 2 = 12
⇒ x = 2 ⇒y = 5

So, (2, 5) is the required solution of the pair of linear equations

1. Elimination method

In this method, we solve the equations by eliminating any one of the variables.

Step 1: Multiply both the equations by such a number so that the coefficient of any one variable becomes equal.

Step 2: Now add or subtract the equations so that the one variable will get eliminated as the coefficients of one variable are same.

Step 3: Solve the equation in that leftover variable to find its value.

Step 4: Substitute the calculated value of variable in the given equations to find the value of the other variable.

Consider x + 2y = 8 and 2x – 3y = 2

Step 1: Make the coefficients of any variable the same by multiplying the equations with constants. Multiplying the first equation by 2, we get,

2x + 4y = 16

Step 2: Add or subtract the equations to eliminate one variable, giving a single variable equation.
Subtract second equation from the previous equation Step 3: Solve for one variable and substitute this in any equation to get the other variable.

y = 2,

x = 8 – 2 y

⇒ x = 8 – 4

⇒ x = 4

(4, 2) is the solution.

1. Cross multiplication method

Given two equations in the form of and , where ### Comparing the ratios of coefficients of a Linear Equation 1. Equations Reducible to a Pair of Linear Equations in Two Variables

There is some pair of equations which are not linear but can be reduced to the linear form by substitutions.

Given equations We can convert these type of equations in the form of ax + by + c = 0

Let and Now after substitution the equation will be

am + bn = c

It can be  solved by any of the method of solving Linear Equations

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