**Solving Linear Equations Using Substitution Method :**

In this section, we will see, how we solve linear linear equations in two variables using the concept substitution method.

We use the following steps to solve a linear equations.

**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.

**Example 1 :**

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

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

√2 x + √3y = 0 and √3 x - √8 y = 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, solving linear equations using substitution method.

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