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What are they?
• Simply 2 equations– With 2 unknowns– Usually x and y
• To SOLVE the equations means we find values of x and y that– Satisfy BOTH equations [work in]– At same time [simultaneously]
Elimination Method
2x – y = 1
3x + y = 9
We have the same number of y’s in each
If we ADD the equations, the y’s disappear+
5x = 10 Divide both sides by 5
x = 2
AB
Substitute x = 2 in equation A2 x 2 – y = 1
4 – y = 1y = 3
Answerx = 2, y = 3
Elimination Method
5x + y = 17
3x + y = 11
We have the same number of y’s in each
If we SUBTRACT the equations, the y’s disappear

2x = 6 Divide both sides by 2
x = 3
AB
Substitute x = 3 in equation A5 x 3 + y = 17
15 + y = 17y = 2
Answerx = 3, y = 2
Elimination Method
2x + 3y = 9
2x + y = 7
We have the same number of x’s in each
If we SUBTRACT the equations,the x’s disappear

2y = 2 Divide both sides by 2
y = 1
AB
Substitute y = 1 in equation A2x + 3 = 9
2x = 6x = 3
Answerx = 3, y = 1
Elimination Method
4x  3y = 14
2x + 3y = 16
We have the same number of y’s in each
If we ADD the equations,the y’s disappear
+
6x = 30 Divide both sides by 6
x = 5
AB
Substitute x = 5 in equation A20 – 3y = 14
3y = 6y = 2
Answerx = 5, y = 2
Basic steps
• Look at equations
• Same number of x’s or y’s?
• If the sign is different, ADD the equations otherwise subtract tem
• Then have ONE equation
• Solve this
• Substitute answer to get the other
• CHECK by substitution of BOTH answers
What if NOT same number of x’s or y’s?
5x + 2y = 173x + y = 10
x = 3
In B
AB
5 x 3 + 2y = 1715 + 2y = 17
y = 1
Answerx = 3, y = 1
If we multiply A by 2 we get 2y in each
5x + 2y = 176x + 2y = 20
B
A
What if NOT same number of x’s or y’s?
3x + 6y = 214x  2y = 8
+15x = 45
In B
AB
3 x 3 + 6y = 216y = 12
y = 2
Answerx = 3, y = 2
If we multiply A by 3 we get 6y in each
3x + 6y = 2112x  6y = 24
B
A
x = 3
…if multiplying 1 equation doesn’t help?
3x + 7y = 265x + 2y = 24
29y = 58
In B
AB
5x + 2 x 2 = 245x = 20
x = 4
Answerx = 4, y = 2
Multiply A by 5 & B by 3, we get 15x in each
15x + 35y = 13015x + 6y = 72B
A
y = 2
Could multiply A by 2 & Bby 7 to get 14y in each
…if multiplying 1 equation doesn’t help?
3x  2y = 75x + 3y = 37
+19x = 95
In B
AB
5 x 5 + 3y = 373y = 12
y = 4
Answerx = 5, y = 4
Multiply A by 3 & B by 2, we get +6y & 6y
9x – 6y = 2110x + 6y = 74B
A
x = 5
Could multiply A by 5 & Bby 3 to get 15x in each
Substitution Method
Given the following equations :
y = x + 3 (i)y = 2x (ii)
Replace the y in equation (i) with 2x from equation (ii)
2x = x + 3 2x – x = 3
x = 3
Sub. x = 3 into either of the two original equations to find the value of yy = x + 3 (i)y = 3 + 3y = 6The answer is(3, 6)
A tool hire firm offers two ways in which a tool may be hired:•Plan A  $20 a day•Plan B  A payment of $40 then $10 a dayFind the number of days whereby there is no difference in the cost of hiring the tool from Plan A and Plan B.
Substitution Method
Let :y = $ in hiring toolx = no. of days hiring tool
y = 20x (1)y = 40 + 10x (2)
Sub.(1) into (2)
20x = 40 + 10x10x = 40
x = 4, y = 80
Graphical Methodx + y = 6Let x = 0 y = 6Coordinates (0, 6)Let y = 0 x = 6Coordinates (6, 0)
2x + y = 8 Let x = 0 y = 8Coordinates (0, 8) Let y = 0 2x = 8 x = 4Coordinates (4, 0)
1. x + y = 62x + y = 8