Make x the subject of ax + b = cx + d Step by Step

Step by step

The question

Question: make x the subject of ax + b = cx + d.

ax + b = cx + d

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

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

1 Get the x terms on one side

cx is on the right.

To remove it, we take away cx.

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

ax + b − cx = cx + d ~~− cx~~

Careful!

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

cx becomes −cx.

Careful: not ax + b + cx = d ✗

cx and −cx make zero.

Cross them out.

Now we have ax + b − cx = d.

ax + b − cx = d

b has no x.

To remove it from the left, we take away b.

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

ax ~~+ b~~ − cx ~~− b~~ = d − b

b and −b make zero.

Cross them out.

Now both x terms are on the left: ax − cx = d − b.

ax − cx = d − b

2 Take x out

x is in every part.

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

ab + ac = a(b + c)

So x times (a − c) is d − b.

x(a − c) = d − b

3 Remove the (a − c)

(a − c)x means (a − c) times x.

We want just x.

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

We do it on both sides.

~~(a − c)~~x~~(a − c)~~ = d − b(a − c)

(a − c) divided by (a − c) is 1.

So just x is left.

So x = (d − b)/(a − c).

x = d − b(a − c)

4 Check

Let's check with numbers.

We pick a = 2, b = 5, c = 3, d = 4.

Our answer gives x = 1.

x = 4 − 52 − 3 = −1−1 = 1

Now we put all the numbers into the start.

First the left side.

Left: (2)(1) − (3)(1) = 2 − 3 = −1

Now the right side.

Right: 4 − 5 = −1

Both sides are −1.

So our answer is right!

Both sides equal −1

Let's look back.

First: get the x terms on one side.

Then: take x out.

Then: remove the (a − c).

Last: check.

Answer