### Answers − Simultaneous Equations by Elimination

#### 1.   Solve these equations by elimination    6x + y = 30    5x − y = 14

i. Label each equation

 Equation Label 6x + y = 30 (1) 5x − y = 14 (2)

ii. Eliminate the variable y by adding the two equations

 6x + y = 30 (1) 5x − y = 14 (2)
 11x = 44 (1) + (2)

iii. Rearrange to find the value of x

 11x = 44 note (3) ÷ 11 = ÷ 11
 x = 4

Note: (3) inverse of × 11 is ÷ 11

iv. Put the value x = 4 into equation (1)

v. Rearrange to find the value of y

 24 + y = 30 note (4) − 24 = − 24
 y = 6

Note: (4) inverse of + 24 is − 24

vi. Verify by putting the value x = 4 and y = 6 into equation (2)

#### 2.   Solve these equations by elimination    5a + 2b = 54    4a + b = 36

answer: a = 6 and b = 12

#### 3.   Solve these equations by elimination    8x + 3y = 47    3x − y = 24

i. Label each equation

 Equation Label 8x + 3y = 47 (1) 3x − y = 24 (2)

ii. Eliminate the y term, by changing one of the equations so that the y terms equate:

 9x − 3y = 72 note (3) + 8x + 3y = + 47 note (1)
 17x = 119 note (4) ÷ 17 = ÷ 17
 x = 7

Note: (3) = 3 × equation (2)
Note: (4) inverse of × 17 is ÷ 17

iii. Put the value x = 7 into equation (1)

iv. Rearrange to find the value of y

 56 + 3y = 47 note (5) − 56 = − 56
 3y = − 9 note (6) ÷ 3 = ÷ 3
 y = − 3

Note: (5) inverse of + 56 is − 56
Note: (6) inverse of × 3 is ÷ 3

v. Verify put the value x = 7 and y = −3 into equation (2)

#### 4.   Solve these equations by elimination    9x + y = 50    7x − y = 78

answer: x = 8 and y = −22

#### 5.   Solve these equations by elimination    5a + 3b = 18    4a − 5b = 44

i. Label each equation

 Equation Label 5a + 3b = 18 (1) 4a − 5b = 44 (2)

ii. Eliminate the b term, by changing both of the equations so that the b terms equate:

 25a + 15b = 90 note (3) 12a − 15b = 132 note (4)
 37a = 222 (3)+(4) ÷ 37 = ÷ 37 note (5)
 a = 6

Note: (3) = 5 × equation (1)
Note: (4) = 3 × equation (2)
Note: (5) inverse of × 37 is ÷ 37

iii. Put the value a = 6 into equation (1)

iv. Rearrange to find the value of b

 30 + 3b = 18 note (6) − 30 = − 30
 3b = − 12 note (7) ÷ 3 = ÷ 3
 b = − 4

Note: (6) inverse of + 30 is − 30
Note: (7) inverse of × 3 is ÷ 3

v. Verify put the value a = 6 and b = −4 into equation (2)

#### 6.   Solve these equations by elimination    7x + y = 67    5x + y = 45

answer: x = 11 and y = −10

#### 7.   Solve these equations by elimination    11x + y = 90    8x + 7y = 78

i. Label each equation

 Equation Label 11x + y = 90 (1) 8x + 7y = 78 (2)

ii. Eliminate the y term, by changing one of the equations so that the y terms equate:

 77x + 7y = 630 note (3) − 8x − 7y = − 78 note (2)
 69x = 552 note (4) ÷ 69 = ÷ 69
 x = 8

Note: (3) = 7 × equation (1)
Note: (2) subtract equation (2)
Note: (4) inverse of × 69 is ÷ 69

iii. Put the value x = 8 into equation (1)

iv. Rearrange to find the value of y

 88 + y = 90 note (5) − 88 = − 88
 y = 2

Note: (5) inverse of + 88 is − 88

v. Verify put the value x = 8 and y = 2 into equation (2)

#### 8.   Solve these equations by elimination    12x + 4y = 84    7x + 2y = 56

answer: x = 14 and y = −21

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