Simultaneous Equations Worked Examples & Solutions (Overdose)

Elimination is easier for most simultaneous equations. For complex equations involving non linear equations you may need to use/ verify with the substitution method.

Example 1

	4x + y = 13...(1)
	2x + 3y =9...(2)
	|*Using elimination*|
	2(4x + y = 13)
	4(2x + 3y = 9)
	8x + 2y = 26
	-(8x + 12y = 36)
	-10y = -10
	y = 1
	|*Substitute y in (1)*|
	4x +1 = 13
	4x = 12
	x = 12/4 = 3

	|*Solutions *|
	x = 3, y =1

Example 2

	3x - y= 18....(1)
	2x + y = 7...(2)
	|*Method of elimination*|
	2(3x- y = 18)
	3(2x + y = 7)
	6x - 2y = 36
	-(6x + 3y = 21)
	-5y = 36- 21 = 15
	y = 15/-5 = -3
	|*Substitute y in (1)*|
	3x -(-3)= 18
	3x +3 = 18
	3x = 15
	x = 15/3 = 5

	8x + 2y = 13....(1)
	3x + y = 4....(2)
	|*Using elimination*|

	3(8x + 2y = 13)
	8(3x + y = 4)
	24x + 6y = 39
	-(24x + 8y = 32)
	-2y = 7
	y = -7/2 

	Put y in (1)
	8x + 2(-7/2) = 13
	8x -7 = 13
	8x = 20
	x = 20/8 = 5/2
	|*Solutions*|
	x = 5/2, y = -7/2

Verifying with Substitution

	|*Pick an easy equation to make one term a subject. In (2) make y the subject*|
	y = 4 -3x
	Substitute y in (1)
	8x + 2(4 - 3x)= 13
	8x + 8 - 6x = 13
	2x = 13 - 8
	2x = 5
	x = 5/2

	|*Substitute x in (1) or (2)*|
	8(5/2) +2y = 13
	20 + 2y = 13
	2y = -7
	y = -7/2

Example 4

	6x +2y = 19....(1)
	2x + 8y = 43....(2)

	2(6x + 2y = 19)
	6(2x + 8y = 43)
	12x + 4y = 38
	-(12x + 48y = 258)
	-44y = -220
	y = -220/-44 = 5

	|*Substitute 5 in (2) or (1)*|

	2x + 8(5)= 43
	2x = 43 - 40
	x = 3/2

Example 5

	2x + 13y = 36.......(1)
	13x + 2y = 69.......(2)

	13(2x + 13y = 36)
	2(13x + 2y = 69)

	26x + 169y = 468
	-(26x + 4y = 138)
	165y =330
	y = 330/ 165 = 2

	|*Sub y = 2 in (1)*|

	2x + 13(2) = 36
	2x + 26 = 36
	2x = 10
	x = 10/2 = 5
	Solutions
	x = 5, y= 2

Example 6

	7x - y = 15.....(1)
	3x - 2y= 19......(2)

	3(7x - y = 15)
	7(3x -2y = 19)

	21x - 3y = 45
	-(21x - 14y = 133)
	11y = -88
	y = -88/11 = -8

	|*Substitute y = -8 in (1)*|
	7x - (-8) = 15
	7x + 8 = 15
	7x = 7
	x =7/7 = 1

	x =1, y = -8

Example 7

	2x - 5y = 22......(1)
	2y - 3x = -11......(2)
	|*Using elimination*|
	|*Rearrange*|
	2x - 5y = 22
	-3x + 2y = -11

	-3(2x - 5y = 22)
	2(-3x + 2y = -11)
	-6x + 15y = -66
	-(-6x + 4y = -22)
	11y = -66 - (-22)= -44
	y = -44/11 = -4

	|*Substitute y = -4 in (1) or (2)*|
	2x - 5(-4) = 22
	2x + 20 = 22
	2x = 20 - 22
	x = 2/2 = 1

Example 8 (Advanced Non linear equations)

	y = x@2 -1.....(1)
	y = 5 - x.....(2)
	|*Using substitution*|
	|*Make x subject*|
	x = 5 - y
	Substitute x in (1)
	y = (5 - y)_2 - 1
	y = (5 - y)(5 - y) -1
	|*Group like terms*|
	25-5y - 5y -y -y@2- 1 = 0
	-y@2 -11y + 24 = 0
	-y@2 -8y -3y + 24 = 0
	-y(y + 8)- 3(y + 8) = 0
	(y + 8)(-y - 3 ) = 0
	y = -8 or -3

	|*Substitute -8 in (2)*|
	-8 = 5 - x
	x = 5 + 8 = 13
	|*Substitute - 3 in (2)*|
	-3 = 5 -x
	x = 3 + 5 = 8
	|*Solutions*|
	y = -8, x =13 and y= -3, x= 8
	or (13, -8) and (8, -3)

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