## What you need to know

Things to remember:

• Solving linear equations involves finding the value of an unknown, usually $x$. We do this by getting it by
• We can solve for $x$ by doing the opposite of what is done to it.
• When we use a mathematical operation on one side, we use the same one on the other side.
• A fraction is a way of writing a division, so $\dfrac{x}{7}=x\div7$
• $6x=6\times x$

Find the value of $x$ so that $5x=35$

This equation is saying that we multiply $x$ by 5 and that gives us 35. We can therefore think of linear equations as a journey. Here we started with $x$, we multiplied by 5, and this gave us 35.

So, if we want to go backwards, we are going to have to do the opposite. And what is the opposite of multiply by 5? Dividing by 5.

$$35\div5=7$$

$$x=7$$

So, to get $x$ by itself, we have to do the opposite. This is the same with dividing, adding, and subtraction.

What happens to $x$                                 What we need to do

$\times$                                           $\div$

$\div$                                               $\times$

+                                                     –

–                                                     +

Let’s use this to try and find some values of $x$

Multiplying

Find the value of $x$ so that $7x=21$

We need to do the opposite of multiplying by 7, which is dividing by 7

$$7x\div7=21\div7$$

$$x=3$$

Dividing

Find the value of $x$ so that $\dfrac{x}{9}=11$

Remember, fractions are just division questions.

$$\frac{x}{9}=x\div9$$

$$x\div9 =11$$

We need to do the opposite of dividing by 9, which is multiplying by 9

$$x\div9\times9 =11\times9$$

$$x=99$$

Find the value of $x$ so that $x+4=11$

We need to do the opposite of adding 4, which is subtracting 4

$$x+4-4=11-4$$

$$x=7$$

Subtract

Find the value of $x$ so that $x-17=3$

We need to do the opposite of subtracting 17, which is adding 17

$$x-17+17=3+17$$

$$x=20$$

## Example Questions

We need to do the opposite of multiplying by 6, which is dividing by 6

$$6x\div7=36\div6$$

$$x=6$$

We need to do the opposite of adding 11, which is subtracting 11

$$x+11-11=11-11$$

$$x=0$$