Step by step
The question
Question: solve x + y = 7 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.
x + x = 2x.
7 + 3 = 10.
Now we have 2x = 10.
2 Find x
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
10 divided by 2 is 5.
So x = 5.
3 Find y
We put x = 5 into ①.
We want y by itself.
The 5 is in the way.
To remove 5, we take away 5.
We do it on both sides, so they stay equal.
5 and − 5 make zero.
Cross them out.
7 minus 5 is 2.
So y = 2.
So x = 5 and y = 2, both at the same time.
We write it as (5, 2).
On a graph, it is the point where the two lines meet.
4 Check
Let's check in both equations.
We put x = 5 and y = 2 into ①.
We get 7, 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.