    ### 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 of
Solving 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 = 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)

• #### Example 2.   Solve the following two equations by elimination                   3x + 2y = 16                   5x + y = 15

• 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:  