Make x the subject of a = (x + b)/(x - c)

Step by step

The question

Question: make x the subject of a = (x + b)/(x − c).

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 − c), is in the way.

Times (x − c) removes it.

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

(x − c) divided by (x − c) is 1.

Only the top is left.

Now we have a(x − c) = x + b.

2 Open the brackets

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

a times every part of (x − c).

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 ax − ac − x = b.

−ac has no x.

To remove it from the left, we add ac.

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

−ac and ac make zero.

Cross them out.

Now both x terms are on the left: ax − x = b + ac.

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 (a − 1) is b + ac.

5 Remove the (a − 1)

(a − 1)x means (a − 1) times x.

We want just x.

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

We do it on both sides.

(a − 1) divided by (a − 1) is 1.

So just x is left.

So x = (b + ac) over (a − 1).

6 Check

Let's check with numbers.

We pick a = 2, b = 3, c = 4.

Our answer gives x = 11.

Now we put all the numbers into the start.

First the left side.

Now the right side.

Both sides are 11.

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 (a − 1).

Last: check.