# SOLVING SYSTEM OF EQUATIONS BY SUBSTITUTION

The following steps will be useful to solve the systems of linear equations using 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.

## Solved Examples

Example 1 :

Solve :

x + y  =  5

2x + 3y  =  5

Solution :

x + y  =  5 -----(1)

2x + 3y  =  5 -----(2)

Step 1 :

Solve (1) for y.

x + y  =  5

Subtract x from each side.

y  =  5 - x -----(3)

Step 2 :

Substitute (5 - x) for y into (2).

(2)-----> 2x + 3(5 - x)  =  5

2x + 15 - 3x  =  5

15 - x  =  5

Subtract 15 from each side.

-x  =  -10

Multiply each side by (-1).

x  =  10

Step 3 :

Substitute 10 for x into (3).

(3)-----> y  =  5 - 10

y  =  -5

Therefore, the solution is

(x, y)  =  (10, -5)

Example 2 :

Solve :

2x + 5y  =  12

4x - y  =  2

Solution :

2x + 5y  =  12 -----(1)

4x - y  =  2 -----(2)

Step 1 :

Solve (2) for y.

4x - y  =  2

Subtract 4x from each side.

-y  =  2 - 4x

Multiply each side by (-1).

y  =  4x - 2 -----(3)

Step 2 :

Substitute (4x - 2) for y into (1).

(2)-----> 2x + 5(4x - 2)  =  12

2x + 20x - 10  =  12

22x - 10  =  12

22x  =  22

Divide each side by 22.

x  =  1

Step 3 :

Substitute 1 for x into (3).

(3)-----> y  =  4(1) - 2

y  =  4 - 2

y  =  2

Therefore, the solution is

(x, y)  =  (1, 2)

Example 3 :

Solve :

x + y - 4  =  0

2x - 3y - 18  =  0

Solution :

x + y - 4  =  0 -----(1)

2x - 3y - 18  =  0 -----(2)

Step 1 :

Solve (1) for x.

x + y - 4  =  0

x + y  =  4

Subtract y from each side.

x  =  4 - y -----(3)

Step 2 :

Substitute (4 - y) for x into (2).

(2)-----> 2(4 - y) - 3y - 18  =  0

8 - 2y - 3y - 18  =  0

-5y - 10  =  0

-5y  =  10

Divide each side by -5.

y  =  -2

Step 3 :

Substitute -2 for y into (3).

(3)-----> x  =  4 - (-2)

x  =  4 + 2

x  =  6

Therefore, the solution is

(x, y)  =  (6, -2)

Example 4 :

Solve :

3x + 5y  =  - 2

2x - y  =  3

Solution :

3x + 5y  =  - 2 -----(1)

2x - y  =  3 -----(2)

Step 1 :

Solve (2) for y.

2x - y  =  3

Subtract 2x from each side.

-y  =  3 - 2x

Multiply each side by (-1).

y  =  2x - 3 -----(3)

Step 2 :

Substitute (2x - 3) for x into (1).

(2)-----> 3x + 5(2x - 3)  =  -2

3x + 10x - 15  =  -2

13x - 15  =  -2

13x  =  13

Divide each side by 13.

x  =  1

Step 3 :

Substitute 1 for x into (3).

(3)-----> y  =  2(1) - 3

y  =  2 - 3

y  =  -1

Therefore, the solution is

(x, y)  =  (1, -1)

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