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Skillsheet
Solving equations by balancing
Imagine an equation as a pan balance.
x 1 4 5 8
x14
8
If we take 2 from the right-hand pan, it will no longer balance.
x 1 4 ≠ 6
x14
6
If we add 2 to the right-hand pan it will not balance.
x 1 4 ≠ 10
x14
10
It will only balance if we add or subtract the same amount from each pan. For example, subtract 2.
x 1 2 5 6
x12
6
We can find out the value of x by taking all of the numbers from the left-hand pan and the same numbers
from the right-hand pan to balance.
x 5 4
x
4
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Solving equations
Example 1
x2356
x59
Add 3 to both sides:
x23
6
x
9
2x
3
Example 2
2x 5 3
05x13
23 5 x
x 523
Add x to both sides:
x13
Take 3 from both sides:
23
x
2x
6
x
3
x
2
4
x
8
Example 3
2x 5 6
x53
Halve both sides
(or divide both sides by 2):
Example 4
x
54
2
x58
Double both sides
(multiply both sides by 2):
Exercises
1
Solve each equation.
a x 1 3 5 7
b e x 1 5 5 0
f 3x 5 12
i 2x 5 210
x 1 5 5 12
x
j 5 8
3
c x 2 6 5 13
g 8x 5 48
k x
2 5 5
2
d x 2 3 5 23
h 25x 5 30
x
l 2 5 6
2
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Solving equations requiring more steps
Often we need to continue to balance the equation to find the value of x.
Example 5
2x 1 7 5 17
2x17
17
2x 5 10
Take 7 from both sides:
2x
10
x55
Divide both sides by 2:
x
5
2(x 1 3)
10
Example 6
2(x 1 3) 5 10
x1355
Divide both sides by 2:
x13
5
x52
Take 3 from both sides:
x
2
Example 7
3x 1 2 5 x 1 10
3x 1 2
x 1 10
Take x from both sides:
2x 1 2
10
2x 5 8
Take 2 from both sides:
2x
8
x54
Divide both sides by 2:
x
4
2x 1 2 5 10
Exercises
2
Solve each equation.
a 6x 2 7 5 17
b 5 1 2x 5 17
c 5x 2 3 5 213
e 7 2 2x 5 1
f 4 2 3x 5 22
i j 4x 1 5 5 2x 1 13
x
g 1 4 5 8
4
k 5x 1 2 5 x 1 10
n 5x 2 2 5 7x 2 12
o 6x 1 13 5 4x 1 13
x
2 3 5 27
2
m x 2 3 5 4x 2 9
d 6 2 3x 5 9
x
2 4 5 27
5
l x 1 6 5 6x 2 9
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Regrouping and balancing
Sometimes it is easier to change the grouping of the terms before balancing the equation.
Example 8
11
2(x 2 4)1(x 1 4)
2(x 2 4) 1 (x 1 4) 5 11
2x 2 8 1 x 1 4 5 11
3x 2 4
11
Add 4 to both sides:
3x
15
Divide both sides by 3:
x
5
3x 2 4 5 11
3x 5 15
x55
Exercises
3 Solve each equation.
a 2(2x 1 3) 1 2(x 1 4) 5 20
c 3(3x 2
4) 2 2(2x 2 3) 5 211
e 2(x 2 7) 5 6(x 1 1)
b 3(x 1 3) 1 2(x 2 5) 5 4
d 3(x 1 4) 5 2(4x 1 1)
f 4(x 1 3) 5 22(x 1 6)
Example 9
2x
2 6 5 22
5
2x
26
5
22
2x
5
4
Multiply both sides by 5:
2x
20
Divide both sides by 2:
x
10
Add 6 to both sides:
2x
54
5
2x 5 20
x 5 10
Exercises
4 'Solve each equation.'
2x
2 6 5 2
5
x12
e 5 2
4
3x 1 1
i 5 10
4
2 2 5x
m 5 23
6
a 3x
1 3 5 15
4
x 15
f 5 1
2
2 x 14
j 5 4
5
7 2 4x
n 53
9
b 4x
1 5 5 1
9
x24
g 523
8
8x 2 5
k 523
7
c 4x
2 11 5 9
3
x 27
522
h 5
3x 2 4
l 52
4
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Answers
1 a x 5 4
e x 5 25
i x 5 10
2 a x 5 4
e x 5 3
i x 5 28
m x 5 2
3 a x 5 1
e x 5 25
4 a x 5 20
e x 5 6
i x 5 13
m x 5 4
b x 5 7
f x 5 4
j x 5 24
b x 5 6
f x 5 2
j x 5 4
n x 5 5
b x 5 1
f x 5 24
b x 5 16
f x 5 23
j x 5 8
n x 5 25
c x 5 19
g x 5 6
k x 5 210
c x 5 22
g x 5 16
k x 5 2
o x 5 0
c x 5 21
d x 5 0
h x 5 26
l x 5 212
d x 5 21
h x 5 215
l x 5 3
c x 5 29
g x 5 220
k x 5 22
d x 5 15
h x 5 23
l x 5 4
d x52
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