    ### Answers − Simultaneous Equations by Substitution

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

i. Label each equation

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

ii. Rearrange equation (1) to make y the subject

 6x + y = 30 note (3) − 6x = − 6x
 y = 30 − 6x

Note: (3) inverse of + 6x is − 6x

iii. Substitute y = 30 − 6x into equation (2):

 5x − y = 14 5x − (30 − 6x) = 14 note (4) 5x − 30  +  6x = 14

Note: (4) − ( − 6) = + 6

iv. Rearrange to find the value of x

 5x − 30 + 6x = 14 note (5) + 30 = + 30
 11x = 44 note (6) ÷ 11 = ÷ 11
 x = 4

Note: (5) inverse of − 30 is + 30
Note: (6) inverse of × 11 is ÷ 11

v. put x = 4 int0:

So x = 4 and y = 6

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

answer: a = 6 and b = 12

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

i. Label each equation

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

ii. Rearrange equation (2) to make y the subject

 3x − y = 24 note (3) + y = + y
 3x = 24 + y note (4) − 24 = − 24
 3x − 24 = + y

Note: (3) inverse of − y is + y
Note: (4) inverse of + 24 is − 24

 So y = 3x − 24

iii. Substitute y = 3x − 24 into equation (1):

 8x + 3(y) = 47 8x + 3(3x − 24) = 47 8x + 9x  −  72 = 47

iv. Rearrange to find the value of x

 8x + 9x − 72 = 47 note (5) + 72 = + 72
 17x = 119 note (6) ÷ 17 = ÷ 17
 x = 7

Note: (5) inverse of − 72 is + 72
Note: (6) inverse of × 17 is ÷ 17

v. put x = 7 into:

So x = 7 and y = −3

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

answer: x = 8 and y = −22

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

i. Label each equation

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

ii. Rearrange equation (1) to make b the subject

 5a + 3b = 18 note (3) − 5a = − 5a
 3b = (18 − 5a) note (4) ÷ 3 = ÷ 3
 So b = Note: (3) inverse of + 5a is − 5a
Note: (4) inverse of × 3 is ÷ 3

 iii. Put b = into (2):
 4a −5 ( b ) = 44 4a −5 ( ) = 44 note(5)
 12a − 5(18 − 5a) = 132 note(6) 12a − 90 + 25a = 132 note(7)

Note: (5) inverse of × 3 is ÷ 3
Note: (6) multiply both sides by 3
Note: (7) expand the brackets by multiplying

iv. Rearrange to find the value of a

 12a − 90 + 25a = 132 note (8) + 90 = +90
 37a = 222 note (9) ÷ 37 = ÷37
 a = 6

Note: (8) inverse of − 90 is + 90
Note: (9) inverse of × 37 is ÷ 37

v. Put a = 6 into:

So a = 6 and b = −4

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

answer: x = 11 and y = −10

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

i. Label each equation

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

ii. Rearrange equation (1) to make y the subject

 11x + y = 90 note (3) − 11x = − 11x
 y = 90 −11x

Note: (3) inverse of + 11x is − 11x

iii. Substitute y = 90 − 11x into equation (2):

 8x + 7(y) = 78 8x + 7(90 − 11x) = 78 8x + 630  −  77x = 78

iv. Rearrange to find the value of x

 8x + 630 − 77x = 78 note (4) − 630 = −630
 −69x = −552 note (5) ÷69 = ÷69
 x = 8

Note: (4) inverse of + 630 − 630
Note: (5) inverse of × 69 is ÷ 69

v. Put x = 8 into:

So x = 8 and y = 2

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

answer: x = 14 and y = −21 back to:   