Solving Simultaneous Equations

1
Solving Simultaneous Equations

• Algebraically

2
What is to be learned

• How to solve simultaneous equations
• by using algebra

3

• 2x 2 8
• x 3
• 2x y 8
• x 3
• y 2
• or x 4,
• x 2,
• etc
• Need More Information

y 0
y 4
4

• 2x y 8
• and x y 5
• x 3, y 2
• Only solution that fits both equations

5
Apples and Bananas

• 4 apples and 3 bananas cost 81p
• 2 apples and 2 bananas cost 46p
• Cost of 6 apples and 5 bananas?
• 127p (81 46)
• Cost of 2 apples and 1 banana
• 35p (81 46)

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As Equations

• 4a 3b 81
• 2a 2b 46
• Adding 6a 5b 127
• 4a 3b 81
• 2a 2b 46
• Subtracting 2a b 35
• You can add or subtract equations

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Some Revision!

• 7 (-5)
• 7 5
• 6b (-4b)
• 6b 4b
• 2b
• 5r (-5r)
• 5r 5r
• 0r
• ? Zero ?

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Solving Simultaneous Equations

• The Elimination Method
• 3a 2b 19
• a 2b 1
• Need to eliminate a or b
• To do so Add or Subtract the equations

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Adding

• a 2b 1
• Add 4a
• b has been eliminated!
• ? a 5
• Need to find value of b
• 15 2b 19
• 2b 4

3 2b 19
a
20
0b
X5
b 2
10

• 5t 2r 22
• 2t 2r 6
• Add 7t 28
• t 4
• 5t 2r 22
• ? 5X4 2r 22
• ? 20 2r 22
• ? 2r 2
• r 1
• t4, r1

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Solving Simultaneous EquationsAlgebraically

• The Elimination Method
• Will have two different letters
• Must eliminate one of them

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Elimination by Adding

• Ex 2x 3y 19
• 3x 3y 6
• Add 5x 25
• x 5
• Sub x 5 into equation
• ? 2X5 3y 19
• 10 3y 19
• 3y 9
• y 3
• x 5, y 3

ys eliminated
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You Try

• 1. x y 8
• x y 2
• 2. 2x 3y 13
• 5x 3y 1
• 3. 4x 2y 16
• 3x 2y 5
• 4. 5x 3y 2
• 2x 3y 5

x 5, y 3 x 2 , y 3
x 3, y 2 x 1 , y -1
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Subtracting

• 4y 3d 17
• 2y 3d 13
• Subtract 2y 4
• y 2
• ? 4X2 3d 17
• ? 8 3d 17
• 3d 9
• d 3
• Solution y 2, d 3

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You Try

• 1. 3x y 8
• x y 4
• 2. 5x 3y 17
• 2x 3y 14
• 3. 4x 2y 18
• 3x 2y 14
• 4. 5x 3y 9
• 3x 3y 3

x 2, y 2 x 1 , y 4
x 4, y 1 x 3 , y -2
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Adding or Subtracting?

• 2m 3t 18 4f 3p 10
• 8m 3t 2 2f 3p 8
• Add 10m 20
• Same Sign ? Subtract

Subtract 2f 2
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Solving Simultaneous EquationsAlgebraically
(continued)

• Can add or subtract the equations
• Same Sign ?
• 2m 3t 7 4f 3p 10
• 8m 3t 2 2f 3p 8

subtract
Subtract
Add
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A New Problem!

• 5t 4r 22
• 3t 2r 12
• ? Same Sign ? Subtract
• BUT 2t 2r 10
• ?
• Can only eliminate if the values are the same
• (same number in front of letter)

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More Apples and Bananas

• 2 apples and 3 bananas cost 57p
• Cost of 4 apples and 6 bananas?
• 114 p (2 X 57)
• We are allowed to multiply!

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Multiplying to Eliminate

• 3t 2r 12
• 5t 4r 22
• Must make the r values equal

X2 ? 6t 4r 24
5t 4r 22
subtract t 2
then continue as before
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And Finally

• Usually you need to multiply both equations
• 3s 2r 17
• 2s - 5r 5

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And Finally

• Usually you need to multiply both equations
• 3s 2r 17
• 2s - 5r 5

X5?
15s 10r 85
X2?
4s - 10r 10
Add 19s 95
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Solving Simultaneous EquationsAlgebraically
(last bit)

• To eliminate a letter the amounts must be equal
  in both equations
• Multiply one (or both) of equations before
• eliminating

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• Ex
• 3s - 3r 9
• 7s - 4r 18

X4?
12s - 12r 36
X3?
21s - 12r 54
Subtract 9s 18 (bottom top)
s 2 etc
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Constructing Simultaneous Equations

length 92cm
4b 3s 92
length 1.4m
6b 5s
1.4
140
b 20, s 4
68cm
length?
26

• Group of adults and children go to theatre
• 20 in group.
• Adults cost 5 and children 3
• Total cost is 76
• How many adults in group?

a adults c children
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• Group of adults and children go to theatre
• 20 in group.
• Adults cost 5 and children 3
• Total cost is 76
• How many adults in group?
• a c 20

a adults c children
28

• Group of adults and children go to theatre
• 20 in group.
• Adults cost 5 and children 3
• Total cost is 76
• How many adults in group?
• a c 20

a adults c children
10 adults
? 10X5
10 children
? 10X3
a adults
? aX5
c children
? cX3
children cost 3c
adults cost 5a
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• Group of adults and children go to theatre
• 20 in group.
• Adults cost 5 and children 3
• Total cost is 76
• How many adults in group?
• a c 20
• 5a 3c 76

a adults c children
children cost 3c
adults cost 5a
8 adults 12 children