Step by step
The question
Question: solve 2x + y = 9 and x − y = 3.
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.
2x + x = 3x.
9 + 3 = 12.
Now we have 3x = 12.
2 Find x
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
12 divided by 3 is 4.
So x = 4.
3 Find y
We put x = 4 into ①.
2 times 4 is 8.
Now we have 8 + y = 9.
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.
9 minus 8 is 1.
So y = 1.
So x = 4 and y = 1, both at the same time.
We write it as (4, 1).
On a graph, it is the point where the two lines meet.
4 Check
Let's check in both equations.
We put x = 4 and y = 1 into ①.
We get 9, the same as the right side.
Now ②.
We get 3, 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.