Introduction to Algebra - Multiplication

Please read Introduction to Algebra first

A Puzzle

What is the missing number?

 
× 4 = 8

The answer is 2, right? Because 2 × 4 = 8.

Well, in Algebra we don't use blank boxes, we use a letter. So we might write:

x × 4 = 8

But the "x" looks like the "×"! ... that could be very confusing ... so in Algebra we don't use the multiply symbol (×) between numbers and letters, we simply put the number next to the letter to mean multiply:

4x = 8

You would say in English "four x equals eight", meaning that 4 x's make 8. And the answer would be written:

x = 2

How to Solve

On the previous page we showed this neat step-by-step approach:

  • Work out what to remove to get "x = ..."
  • Remove it by doing the opposite
  • Do that to both sides

It still works, but you have to know that dividing is the opposite of multiplying. Have a look at this example:

We want to
remove the "4"
To remove it, do
the opposite
, in
this case divide by 4:
Do it to both sides: Which is ... Solved!

Why did we divide by 4 on both sides?

Because of the need for balance ...

  Divide Left by 4 Divide Right by 4 Also
In Balance Out of Balance! In Balance Again

Just remember ...

To keep the balance, what you do to one side of the "="
you should also do to the other side!

Another Puzzle

Solve this one:
x / 3 = 5

Start with: x/3 = 5
   

What you are aiming for is an answer like "x = ...", and the divide by 3 is in the way of that!

If you multiply by 3 you can cancel out the divide by 3 (because 3/3=1)

 
   
So, let us have a go at multiplying by 3 on both sides: x/3 ×3 = 5 ×3
   
A little arithmetic (3/3 = 1 and 5×3 = 15) becomes: 1x = 15
   
Which is just: x = 15
  Solved!
(Quick Check: 15/3 = 5)  

Have a Try Yourself

Now practice on this Algebra Multiplication Worksheet and then check your answers on the page after. Try to use the steps we have shown you here, rather than just guessing!

More Complicated Example

How would you solve this?

x / 3 + 2 = 5

It might look hard, but not if you solve it in stages.

First let us get rid of the "+2":

Start with: x/3 + 2 = 5
   

To remove the plus 2 use minus 2 (because 2-2=0)

x/3 + 2 -2 = 5 -2
   
A little arithmetic (2-2 = 0 and 5-2 = 3) becomes: x/3 + 0 = 3
   
Which is just: x/3 = 3

Now, get rid of the "/3":

Start with: x/3 = 3
   

If you multiply by 3 you can cancel out the divide by 3:

x/3 ×3 = 3 ×3
   
A little arithmetic (3/3 = 1 and 3×3 = 9) becomes: 1x = 9
   
Which is just: x = 9
  Solved!
(Quick Check: 9/3 + 2 = 3+2 = 5)  

When you get more experienced:

When you get more experienced, you can solve it like this:

Start with: x/3 + 2 = 5

Subtract 2 from both sides:

x/3 + 2 -2 = 5 -2
Simplify: x/3 = 3

Multiply by 3 on both sides:

x/3 ×3 = 3 ×3
Simplify: x = 9
Or even like this:
Start with: x/3 + 2 = 5

Subtract 2:

x/3 = 3

Multiply by 3:

x = 9

 


Real World Example

Example: Sam bought 3 boxes of chocolates online.
Postage was $9 and the total cost was $45.
How much was each box?

 

Let's use x for the price of each box.

3 times x plus $9 is $45:

3x + 9 = 45

Let's solve!

Start with: 3x + 9 = 45
Subtract 9 from both sides: 3x + 9 − 9 = 45 − 9
Simplify: 3x = 36
Divide by 3: 3x /3 = 36 /3
Simplify: x = 12

So each box was $12

Advanced: you can also do the "divide by 3" first (but you must do it to all terms):

Start with: 3x + 9 = 45
Divide by 3: 3x/3 + 9/3 = 45/3
Simplify: x + 3 = 15
Subtract 3 from both sides: x + 3 − 3 = 15 − 3
Simplify: x = 12

Have a Try Yourself

Now practice on this Algebra (Two Steps to Solve) Worksheet and then check your answers on the page after. Try to use the steps we have shown you here, rather than just guessing!