Step by step
The question
Question: solve 4x − y = 6 and x + y = 4.
We call them ① and ②.
We want the x and y that make both true.
We will add the equations to remove one letter.
This way is called elimination.
1 Add the equations
Look at y: −y and y.
They are opposites.
Added, they make zero.
Why can we add equations?
Both sides of ② are the same amount.
So we add the same amount to both sides of ①.
−y and y make zero.
Cross them out.
We add ① and ②: the x parts together, and the numbers together.
4x + x = 5x.
6 + 4 = 10.
Now we have 5x = 10.
2 Find x
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just x is left.
10 divided by 5 is 2.
So x = 2.
3 Find y
We put x = 2 into ①.
4 times 2 is 8.
Now we have 8 − y = 6.
We want y by itself.
The 8 is in the way.
To remove 8, we take away 8.
We do it on both sides, so they stay equal.
8 and − 8 make zero.
Cross them out.
6 minus 8 is −2.
Now we have −y = −2.
−y means −1 times y.
We want just y.
To remove times −1, we divide by −1.
We do it on both sides, so they stay equal.
−1 divided by −1 is 1.
So just y is left.
−2 divided by −1 is 2.
So y = 2.
So x = 2 and y = 2, both at the same time.
We write it as (2, 2).
On a graph, it is the point where the two lines meet.
4 Check
Let's check in both equations.
We put x = 2 and y = 2 into ①.
We get 6, the same as the right side.
Now ②.
We get 4, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: add the equations.
Then: find x.
Then: find y.
Last: check.