1. Simultaneous Equations 6-Apr-17
Objective
Solve a pairs of simultaneous equations like’
3x + 2y = 20
4x – 4y = 10
and
3x + 2y = 36
2x + 3y = 29
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2. Here are some of the answers
For an equation like x + 3 = 5, there is only one unknown value and this is x = 2
But an equation like x + y = 5 with two unknown values, has lots of (infinite) answers
Here are some of the answers
x + y = 5
x = 4
y = 1
4 + 1 = 5
x = 3
y = 2
3 + 2 = 5
x = 2
y = 3
2 + 3 = 5
x = 1
y = 4
1 + 4 = 5
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3. Pairs of equations like this are called simultaneous equations
But if we add a second equation like this:
x + y = 5
x – y = 1
there is only one value of x and one value of y that will fit into two equations
3 + 2 = 5
3 – 2 = 1
x = 3 and y = 1
Pairs of equations like this are called simultaneous equations
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4. Middle signs are different so
3x + 2y = 16 (1)
2x - 2y = 4 (2)
Middle signs are different so
(1) + (2) will eliminate y
First make sure that the middle numbers are the same and both are 2y, so this is OK.
Look at the signs in front of the middle numbers.
Same SUBTRACT Different ADD.
Finally, replace the x with 4 in one of the equations to find y
It’s a good idea to label the equations
(1) and (2)
5x = 20
x = 20 ÷ 5
x = 4
3x + 2y = 16
(3x4) + 2y = 16
12 + 2y = 16
2y =
2y = 4
y = 2
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5. Middle signs are the same so (1) - (2) will eliminate y
2. 4x + 3y = 32 (1)
x + 3y = 17 (2)
Middle signs are the same so
(1) - (2) will eliminate y
First make sure that the middle numbers are the same and both are 3y, so this is OK.
Look at the signs in front of the middle numbers.
Same SUBTRACT Different ADD.
Finally, replace the x with 5 in one of the equations to find y
It’s a good idea to label the equations
(1) and (2)
3x = 15
x = 15 ÷ 3
x = 5
x + 3y = 17
5 +3y = 17
3y = 12
y = 4
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6. 1. 4x + 3y = 32
x + 3y = 17
2. 3x - 2y = 17
2x + 2y = 28
3. 4x + y = 41
5x – y = 49
4. 5x - 3y = 18
3x – 3y = 6
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7. Simultaneous Equations with Different Numbers in the Middles
4x + 3y = 15
5x – 2y = 13
To solve the equations, these numbers need to be the same. The following slides explain how you can do this.
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8. 4x + 3y = 15 Multiply this line by 2
(2 x 4x) + (2 x 3y) = (2 x 15)
(3 x 5x) – (3 x 2y) = (3 x 13)
8x + 6y = 30
15x – 6y = 39
Now, the middle numbers are the same.
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9. 8x + 6y = 30
15x – 6y = 39
The middles are the same
If the signs are different ADD
Find the value of the remaining letter
23x = 69
x = 69 ÷ 23
x = 3
8x + 6y = 30
(8 x 3) + 6y = 30
24 + 6y = 30
3y = 30 – 24
3y = 6
y = 2
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10. The middles need to be the same numbers
5x - 2y = 20
3x – 3y = 3
5x - 2y = 20 Multiply this line by 3
3x – 3y = 3 Multiply this line by 2
(3 x 5x) - (3 x 2y) = (3 x 20)
(2 x 3x) – (2 x 3y) = (2 x 3)
15x - 6y = 60
6x – 6y = 6
Now, the middle numbers are the same
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11. 15x - 6y = 60
6x – 6y = 6
The middles are the same
If the signs are the same SUBTRACT
Find the value of the remaining letter
9x = 54
x = 54 ÷ 9
x = 6
3x - 3y = 3
(3 x 6) - 3y = 3
18 - 3y = 3
3y = 18 – 3
3y = 15
y = 5
© Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09