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Solving Simple Equations (2 of 3)

By Subject > Algebra > Simple Linear Equations > Solving Equations (Multiplication & Division)

Equations With Multiplication and Division

We can use the same "do the same thing to both sides" idea that we use to solve equations with addition and subtraction to solve equations that involve multiplication and division.

Multiplication and Division Recap

When we multiply we combine groups of equal numbers .....
o o o o
o o o o
o o o o

3 groups of 4

3 x 4

o o o o o o
o o o o o o

3 x 4 = 12

 

... and when we divide we split into equal groups
o o o o o
o o o o o

10
o o o o o o
o o o o o o

10 divided into
2 groups of 5

10 ÷ 2 = 5

We use these operations and the fact that they are opposite operations to help us solve equations.
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Dividing Both Sides By The Same Number

We can solve the equations below by "getting rid of" or "cancelling out" the multiplication by dividing (doing the opposite operation.)

Note: There is a short way to write multiplication.
Instead of 3 x n
we can write just 3n
(we can leave out the x sign)

Equation
 Left Right
3n = 12
n
n
n
o o o o o o
o o o o o o

Divide both sides by 3

3n ÷ 3 = 12 ÷ 3

n
n
n
o o o o
o o o o
o o o o
n = 4
n
o o o o

Here is a similar equation.

Equation
 Left Right
5n = 15
n
n
n
n
n
o o o o o o o o
o o o o o o o

Divide both sides by 5

5n ÷ 5 = 15 ÷ 5

n
n
n
n
n
o o o
o o o
o o o
o o o
o o o
n = 3
n
o o o

Multiplying Both Sides By The Same Number

We can solve the equations below by "getting rid of" or "cancelling out" the division by multiplying (doing the opposite operation.)

Note: There are different ways to write division
Instead of 3 ÷ n
we can write
3
n

Equation
 Left Right
n = 2
3

 

 n
o o

Multiply both sides by 3

n x 3 = 2 x 3
3

 

 n
o o
o o
o o
n = 6
n
o o o o o o

The same idea is used to solve the equation below:

Equation
 Left Right
n = 4
5

 

 n
o o o o

Multiply both sides by 5

n
x 5 = 4 x 5
5

 

 n
o o o o
o o o o
o o o o
o o o o
o o o o
n = 20
n
o o o o o o o o o o o o o o o o o o o o

You can "get rid of" or "cancel" multiplication by dividing and you can "get rid of" or "cancel" division by multiplying.

More Examples of Solving Equations

Problem
Solution
3n = 18
 
Divide both sides by 3

3n ÷ 3 = 18 ÷ 3

1n = 6

n = 6

 

Problem
Solution
n
= 2
6
 
Multiply both sides by 6
n
x 6 = 2 x 6
6
n
= 12
1

n = 6

 

Problem
Solution
9n = 72
 
Divide both sides by 9

9n ÷ 9 = 72 ÷ 9

n = 8

 

Problem
Solution
a
= 11
7
 
Multiply both sides by 7
n
x 7 = 11 x 7
7

a = 77

Try the printable worksheets below to practice solving equations through multiplication and division.

The two worksheets below combine the use of addition, subtraction, multiplication, and division to solve equations.

The worksheets listed above are included here with more solving equations worksheets.



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