**Solving Linear Equations in Two Variables by Substitution :**

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.

(i) x + y = 14 and x - y = 4

**Solution :**

x + y = 14 ------(1)

x - y = 4------(2)

**Step 1 :**

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

y = 14 - x

**Step 2 :**

Now we have to substitute the value of y in the other equation and reduce it to one equation of one variable.

By applying the value of y in (2).

x - (14 - x) = 4

x - 14 + x = 4

2x - 14 = 4

Add 14 on both sides

2x = 4 + 14

2x = 18

Divide 2 on both sides

x = 18/2

x = 9

**Step 3 :**

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

y = 14 - x

y = 14 - 9

y = 5

So, the solution is (9, 5).

(ii) s - t = 3 and (s/3) + (t/2) = 6

**Solution :**

**s - t = 3 ----(1)**

**(s/3) + (t/2) = 6 -----(2)**

**Step 1 :**

Find the value of one variable in terms of other variable, say s in terms of t

s = 3 + t

**Step 2 :**

Now we have to substitute the value of s in the other equation and reduce it to an equation of one variable.

By simplifying the second equation, we get

(2s + 3t)/6 = 6

2s + 3t = 36

By applying the value of s, we get

2s + 3t = 36

2(3 + t) + 3t = 36

6 + 2t + 3t = 36

6 + 5 t = 36

5t = 36 - 6

5t = 30

t = 30/5

t = 6

**Step 3 :**

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

s = 3 + t

s = 3 + 6

s = 9

So, solution is (9, 6).

(iii) 3x - y = 3 and 9x - 3y = 9

**Solution :**

**3x - y = 3 -----(1)**

**9x - 3y = 9 ----(2)**

**Step 1 :**

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

y = 3 x - 3

**Step 2 :**

Now we have to substitute the value of y in the other equation and reduce it to an equation of one variable.

9x - 3(3x - 3) = 9

9x - 9x + 9 = 9

The statement is true, from this we can decide that the pair of linear equations has infinitely many solution.

After having gone through the stuff given above, we hope that the students would have understood, solving linear equations in two variables by substitution.

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