MATHEMATICS; INTRODUCTION TO ALGEBRA (2)

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 can be very confusing ... so in Algebra we don't use the multiply symbol (×) between numbers and letters:

We put the number next to the letter to mean multiply:

4x

=

8

In English we say "four x equals eight", meaning that 4 x's make 8.

 

And the answer is written:

x

=

2

How to Solve

Instead of saying "obviously x=2", use 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

And what is the opposite of multiplying? Dividing!

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 ...

In Balance

 

Divide Left by 4

Out of Balance!

 

Divide Right by 4 Also

In Balance Again

 

Just remember ...

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

Another Puzzle

Solve this one:

x

/

3

=

5

What we want is an answer like "x = ...", but the divide by 3 is in the way of that!

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

So, let us try multiplying by 3 on both sides:x3 ×3 = 5 ×3

A little arithmetic (13×3 = 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 do we solve this?

x

/

3

+

2

=

5

It might look hard, but not if we 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

multiply by 3 to 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 faster like this:

Start with:x/3 + 2 = 5

Subtract 2 from both sides: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: we can also do the "divide by 3" first (but we 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

Same answer.

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!