Make x the subject of px + q = rx Step by Step

Step by step

The question

Question: make x the subject of px + q = rx.

px + q = rx

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

rx is on the right.

To remove it, we take away rx.

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

px + q − rx = ~~rx − rx~~

Careful!

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

rx becomes −rx.

Careful: not px + q + rx = 0 ✗

rx and −rx make zero.

Cross them out.

Now we have px + q − rx = 0.

px + q − rx = 0

q has no x.

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

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

px ~~+ q~~ − rx ~~− q~~ = 0 − q

q and −q make zero.

Cross them out.

Now both x terms are on the left: px − rx = −q.

px − rx = −q

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 (p − r) is −q.

x(p − r) = −q

3 Remove the (p − r)

(p − r)x means (p − r) times x.

We want just x.

To remove times (p − r), we divide by (p − r).

We do it on both sides.

~~(p − r)~~x~~(p − r)~~ = −q(p − r)

(p − r) divided by (p − r) is 1.

So just x is left.

So x = −q/(p − r).

x = −q(p − r)

4 Check

Let's check with numbers.

We pick p = 2, q = 4, r = 3.

Our answer gives x = 4.

x = −42 − 3 = −4−1 = 4

Now we put all the numbers into the start.

First the left side.

Left: (2)(4) − (3)(4) = 8 − 12 = −4

Now the right side.

Right: −4

Both sides are −4.

So our answer is right!

Both sides equal −4

Let's look back.

First: get the x terms on one side.

Then: take x out.

Then: remove the (p − r).

Last: check.

Answer