Solving Simultaneous Equations by Elimination

1. Eliminate one of the variables.

2. Find the value of the remaining variable.

3. Substitute this value into any of the starting equations.

4. State the final answer, giving values for both variables, eg. x = 3, y = 4

5. Check answer by substituting the values into one of the original equations.

##### Solve:

3x + 3y = 27                 (eq. 1)

3x − y = 3                     (eq. 2)

Eliminate one variable: the 2 variables are x and y. Look for a variable that is the same in both equations (in this case, 3x appears in both). If we subtract (eq. 2) from (eq. 1)…

3x – 3x = 0

3y – (– y) = 3y + y = 4y

27 - 3 = 24

We end up with 4y = 24

Find the value of the remaining variable

4y = 24

divide both sides by 4

y = 6

Substitute this value into any of the equations

3x − y = 3

3x − 6 = 3                     Add 6 to both sides

3x = 9                          Divide both sides by 4

x = 9/3 = 3

State the final answer, giving values for both variables

x = 3, y = 6

3x + 3y = 27

3(3) + 3(6)

9 + 18 = 27                  Answer is correct

##### Solve:

6x + 4y = 16                             (1)

6x + 15y = -6                           (2)

Take (2) from (1) to remove x:

6x – 6x = 0

4y – 15y = – 11y

16 – (–6) = 22

-11y = 22                     Divide both sides by -11

y = -2

Substitute for y           6x + 4y = 16

6x + 4(-2) = 16

6x - 8 = 16                   Add 8 to both sides

6x = 24                        Divide both sides by 6

x = 4

x = 4, y = -2                  Check: we’ll use equation (2)…          6x + 15y = -6

6(4) + 15(-2)

24 – 30 = -6

##### What happens when the numbers before variables don’t match?

6x + 5y = 13     (1)

2x + 3y = 3       (2)

These are equations – one side equals the other. We can alter them and if we perform the same operation to every term, the equation remains the same!

We could change the 2x in (2) to 6x if we multiply everything by 3…

2x + 3y = 3       becomes         6x + 9y = 9       (3)

Now take (1) away from (3)

6x + 9y = 9

6x + 5y = 13

4y = –4

y = –1                substitute into (2)      2x + 3(–1) = 3

2x – 3 = 3         add 3 to both sides

2x = 6              divide both sides by 2

x = 3

Our answer is…           x = 3, y = –1