Simultaneous Equations by Elimination
Simultaneous equations are two equations with two unknown variables, and we want to find the same solution
In steps that rearrange the equation we use a combination ofSolving Equations − Add and Subtract and Solving Equations − Multiply and Divide
Please look at the above before continuing
Example 1. Solve the following two equations by elimination
6x + 3y = 30
9x − 3y = 15Example 2. Solve the following two equations by elimination
3x + 2y = 16
5x + y = 15
i. Label each equation
| Equation | Label | ||||
| 6x + 3y | = | 30 | (1) | ||
| 9x − 3y | = | 15 | (2) | ||
ii. Eliminate the variable y by adding the two equations
| 6x + 3y | = | 30 | (1) | ||||
| 9x − 3y | = | 15 | (2) |
| 15x | = | 45 | (1) + (2) |
iii. Rearrange to find the value of x.
| 15x | = | 45 | note (3) | ||||
| ÷ | 15 | = | ÷ | 15 |
| x | = | 3 |
Note: (3) inverse of × 15 is ÷ 15
iv. Put the value x = 3 into equation (1)
v. Rearrange to find the value of y.
| 18 + 3y | = | 30 | note (4) | |||
| − | 18 | = | −18 |
| 3y | = | 12 | note (5) | |||
| ÷ | 3 | = | ÷ 3 |
| y | = | 4 |
Note: (4) inverse of + 18 is − 18
Note: (5) inverse of × 3 is ÷ 3
vi. Verify by putting the value x = 3 and y = 4 into equation (2)
i. Label each equation
| Equation | Label | ||||
| 3x + 2y | = | 16 | (1) | ||
| 5x + y | = | 15 | (2) | ||
ii. Eliminate the y term, by changing one of the equations so that the y terms equate.
| 10x + 2y | = | 30 | note (3) | |||
| −3x − 2y | = | −16 | note (1) |
| 7x | = | 14 | note (4) | |||||
| ÷ | 7 | = | ÷ | 7 |
| x | = | 2 |
Note: (3) = 2 × equation (2)
Note: (1) subtract equation (1)
Note: (4) inverse of × 7 is ÷ 7
iii. Put the value x = 2 into equation (1)
iv. Rearrange to find the value of y.
| 6 + 2y | = | 16 | note (5) | ||||
| − | 6 | = | − | 6 |
| 2y | = | 10 | note (6) | ||||
| ÷ | 2 | = | ÷ | 2 |
| y | = | 5 |
Note: (5) inverse of + 6 is − 6
Note: (6) inverse of × 2 is ÷ 2
v. Verify put the value x = 2 and y = 5 into equation (2)
to: 
