SOLVING EQUATIONS BY SUBSTITUTION METHOD

The following steps will be useful to solve system of linear equations using method of substitution.

Step 1 : 

In the given two equations, solve one of the equations either for x or y. 

Step 2 : 

Substitute the result of step 1 into other equation and solve for the second variable. 

Step 3 : 

Using the result of step 2 and step 1, solve for the first variable.  

Examples

Example 1 : 

Solve for x and y : 

x - 5y + 17  =  0

2x + y + 1  =  0

Solution : 

x - 5y + 17  =  0 -----(1)

2x + y + 1  =  0 -----(2)

Step 1 :

Solve (1) for x. 

x - 5y + 17  =  0

Subtract 17 from each side. 

x - 5y  =  -17

Add 5y to each side.

x  =  5y - 17 -----(3)

Step 2 : 

Substitute (5y - 17) for x into (2). 

(2)-----> 2(5y - 17) + y + 1  =  0

10y - 34 + y + 1  =  0

11y - 33  =  0

Add 33 to each side.

11y  =  33

Divide each side by 11.

y  =  3

Step 3 :

Substitute 3 for y into (3).

(3)-----> x  =  5(3) - 17

x  =  15 - 17

x  =  -2

Therefore, the solution is 

(x, y)  =  (-2, 3)

Example 2 : 

Solve for x and y : 

5x + 3y - 8  =  0

2x - 3y + 1  =  0

Solution : 

5x + 3y - 8  =  0 -----(1)

2x - 3y + 1  =  0 -----(2)

Step 1 :

Solve (1) for 3y. 

5x + 3y - 8  =  0

Add 8 to each side. 

5x + 3y  =  8

Subtract 5x from each side.

3y  =  8 - 5x -----(3)

Step 2 : 

Substitute (8 - 5x) for 3y into (2). 

(2)-----> 2x - (8 - 5x) + 1  =  0

2x - 8 + 5x + 1  =  0

7x - 7  =  0

Add 7 to each side.

7x  =  7

Divide each side by 7.

x  =  1

Step 3 :

Substitute 1 for x into (3).

(3)-----> 3y  =  8 - 5(1)

3y  =  8 - 5

3y  =  3

Divide each side by 3.

y  =  1

Therefore, the solution is 

(x, y)  =  (1, 1)

Example 3 : 

Solve for x and y : 

4x - 7y  =  0

8x - y - 26  =  0

Solution : 

4x - 7y  =  0 -----(1)

8x - y - 26  =  0 -----(2)

Step 1 :

Solve (1) for 4x. 

4x - 7y  =  0

Add 7y to each side. 

4x  =  7y -----(3)

Step 2 : 

Substitute 7y for 4x into (2). 

(2)-----> 8x - y - 26  =  0

2(4x) - y - 26  =  0

2(7y) - y - 26  =  0

Simplify.

14y - y - 26  =  0

13y - 26  =  0

Add 26 to each side.

13y  =  26

Divide each side by 13.

y  =  2

Step 3 :

Substitute 2 for y into (3).

(3)-----> 4x  =  7(2)

4x  =  14

Divide each side by 4.

x  =  3.5

Therefore, the solution is 

(x, y)  =  (3.5, 2)

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