Step by step
The question
Question: make x the subject of y = (2x + 1)/(x − 4).
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 − 4), is in the way.
Times (x − 4) removes it.
We do it on both sides, so they stay equal.
(x − 4) divided by (x − 4) is 1.
Only the top is left.
Now we have y(x − 4) = 2x + 1.
2 Open the brackets
What is in front of a bracket multiplies every part inside.
y times every part of (x − 4).
3 Get the x terms on one side
2x is on the right.
To remove it, we take away 2x.
We do it on both sides, so they stay equal.
Careful!
A term that moves to the other side changes its sign.
2x becomes −2x.
2x and −2x make zero.
Cross them out.
Now we have xy − 4y − 2x = 1.
−4y has no x.
To remove it from the left, we add 4y.
We do it on both sides, so they stay equal.
−4y and 4y make zero.
Cross them out.
Now both x terms are on the left: xy − 2x = 1 + 4y.
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 − 2) is 1 + 4y.
5 Remove the (y − 2)
(y − 2)x means (y − 2) times x.
We want just x.
To remove times (y − 2), we divide by (y − 2).
We do it on both sides.
(y − 2) divided by (y − 2) is 1.
So just x is left.
So x = (1 + 4y)/(y − 2).
6 Check
Let's check with numbers.
We pick y = 3.
Our answer gives x = 13.
Now we put all the numbers into the start.
First the left side.
Now the right side.
Both sides are 13.
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 − 2).
Last: check.