Step by step
The question
Question: solve 2x + 3y = 12 and x − y = 1.
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 Make the y numbers opposite
We want to remove y.
In ① it is 3y, in ② it is −y.
We make them opposite: 3y and −3y.
Then adding removes y.
We multiply every part of ② by 3.
We do it on both sides, so they stay equal.
3 times x is 3x.
3 times −y is −3y.
3 times 1 is 3.
We call it ③.
2 Add the equations
We write ① and ③ under each other: x under x, y under y.
Look at y: 3y and −3y.
They are opposites.
Added, they make zero.
Why can we add equations?
Both sides of ③ are the same amount.
So we add the same amount to both sides of ①.
3y and −3y make zero.
Cross them out.
We add ① and ③: the x parts together, and the numbers together.
2x + 3x = 5x.
12 + 3 = 15.
Now we have 5x = 15.
3 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.
15 divided by 5 is 3.
So x = 3.
4 Find y
We put x = 3 into ②.
We want y by itself.
The 3 is in the way.
To remove 3, we take away 3.
We do it on both sides, so they stay equal.
3 and − 3 make zero.
Cross them out.
1 minus 3 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 = 3 and y = 2, both at the same time.
We write it as (3, 2).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 3 and y = 2 into ①.
We get 12, the same as the right side.
Now ②.
We get 1, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: make the y numbers opposite.
Then: add the equations.
Then: find x.
Then: find y.
Last: check.