Make x the subject of y = (x + 2)/(x - 3)

Step by step

The question

Question: make x the subject of y = (x + 2)/(x − 3).

The subject is the letter on its own: we want x = ...

.

Here x is in two places, so we must bring them together.

1 Clear the fraction

The bottom, (x − 3), is in the way.

Times (x − 3) removes it.

We do it on both sides, so they stay equal.

(x − 3) divided by (x − 3) is 1.

Only the top is left.

Now we have y(x − 3) = x + 2.

2 Open the brackets

What is in front of a bracket multiplies every part inside.

y times every part of (x − 3).

3 Get the x terms on one side

x is on the right.

To remove it, we take away x.

We do it on both sides, so they stay equal.

Careful!

A term that moves to the other side changes its sign.

x becomes −x.

x and −x make zero.

Cross them out.

Now we have xy − 3y − x = 2.

−3y has no x.

To remove it from the left, we add 3y.

We do it on both sides, so they stay equal.

−3y and 3y make zero.

Cross them out.

Now both x terms are on the left: xy − x = 2 + 3y.

4 Take x out

x is in every part.

So we take it out once, and write what is left in a bracket.

So x times (y − 1) is 2 + 3y.

5 Remove the (y − 1)

(y − 1)x means (y − 1) times x.

We want just x.

To remove times (y − 1), we divide by (y − 1).

We do it on both sides.

(y − 1) divided by (y − 1) is 1.

So just x is left.

So x = (2 + 3y)/(y − 1).

6 Check

Let's check with numbers.

We pick y = 2.

Our answer gives x = 8.

Now we put all the numbers into the start.

First the left side.

Now the right side.

Both sides are 8.

So our answer is right!

Let's look back.

First: clear the fraction.

Then: open the brackets.

Then: get the x terms on one side.

Then: take x out.

Then: remove the (y − 1).

Last: check.