# SOLVING LINEAR EQUATIONS USING SUBSTITUTION METHOD

Solving Linear Equations Using Substitution Method :

In this section, you will learn how to solve linear linear equations in two variables using the substitution method.

## Substitution Method - Steps

Step 1 :

Solve one of the equations for one of its variables.

Step 2 :

Substitute the expression from step 1 into the other equation and solve for the other variable.

Step 3 :

Substitute the value from step 2 into either original equation and solve to find the value of the variable in step 1.

## Substitution Method - Examples

Example 1 :

Solve the following pair of linear equations by the substitution method.

0.2x + 0.3y  =  1.3 and 0.4x + 0.5y  =  2.3

Solution :

0.2 x + 0.3 y = 1.3 ------(1)

0.4 x + 0.5 y = 2.3  ------(2)

Multiply both (1) and (2) by 10,

2 x + 3 y = 13 ----(1)

4 x + 5 y = 23 -----(2)

Step 1 :

Find the value of one variable in terms of other variable, say y in terms of x

3y  =  13 - 2x

y  =  (13 - 2x)/3

Step 2 :

By applying the value of y in the second equation, we get

4 x + 5 [(13 - 2x)/3]  =  23

12 x + [5 (13 - 2 x)]/3  =  23

12 x + 65 - 10 x  =  69

2x  =  69 - 65

2 x  =  4

x  =  2

Step 3 :

Now,we have to apply the value of x in the equation

y  =  (13 -2x)/3

y  =  (13 -2(2))/3

y  =  (13 -4)/3

y  =  9/3

y  =  3

So, the solution is (2, 3).

Example 2 :

Solve the following pair of linear equations by the substitution method.

√2x + √3y  =  0 and √3x - √8y  =  0

Solution :

Step 1 :

Find the value of one variable in terms of other variable, say y in terms of x

√3 y  =  - √2 x

y  =  - (√2/√3) x

Step 2 :

By applying the value of y in the second equation, we get

√3x - √8 [- (√2/√3) x]  =  0

√3x + (√16/√3) x)  =  0

(3x + 4x)/√3  =  0

7x/√3  =  0

7x  =  0

x  =  0

Step 3 :

Now, we have to apply the value of x in the equation

y  =  - (√2/√3) x

y  =  - (√2/√3) (0)

y  =  0

So, the solution is (0, 0).

Example 3 :

Solve the following pair of linear equations by the substitution method.

(3x/2) - (5y/3)  =  -2 and (x/3) + (y/2)  =  13/6

Solution :

(3x/2) - (5y/3)  =  -2  --------(1)

(x/3) + (y/2)  =  13/6   --------(2)

We are going to take L.C.M for both equations.

(9x - 10y)/6  =  -2

9x - 10y  =  -12 ------(1)

(x/3) + (y/2)  =  13/6

(2x + 3y)/6  =  13/6

2x + 3y  =  13  ------(2)

Step 1 :

Find the value of one variable in terms of other variable, say y in terms of x

10 y  =  9x + 12

y  =  (9x + 12)/10

Step 2 :

By applying the value of y in the second equation, we get

2x + 3[(9x + 12)/10]  =  13

(20x + 27x + 36)/10  =  13

47x + 36  =  130

47x  =  130 - 36

47x  =  94

x  =  94/47

x  =  2

Step 3 :

Now,we have to apply the value of x in the equation

y  =  (9 x + 12)/10

y  =  (9(2) + 12)/10

y  =  (18 + 12)/10

y  =  30/10

y  =  3

So, the solution is (2, 3). After having gone through the stuff given above, we hope that the students would have understood how to solve linear equations using substitution method.

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