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: (1) add equation (1)
   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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