Step by step
The question
Question: solve x + 3y = 14 and x + y = 6.
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 x numbers opposite
We want to remove x.
In ① it is x, in ② it is x.
We make them opposite: x and −x.
Then adding removes x.
We multiply every part of ② by −1.
We do it on both sides, so they stay equal.
−1 times x is −x.
−1 times y is −y.
−1 times 6 is −6.
We call it ③.
2 Add the equations
We write ① and ③ under each other: x under x, y under y.
Look at x: x and −x.
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.
x and −x make zero.
Cross them out.
We add ① and ③: the y parts together, and the numbers together.
3y − y = 2y.
14 + (−6) = 8.
Now we have 2y = 8.
3 Find y
2y means 2 times y.
We want just y.
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 y is left.
8 divided by 2 is 4.
So y = 4.
4 Find x
We put y = 4 into ①.
3 times 4 is 12.
Now we have x + 12 = 14.
We want x by itself.
The + 12 is in the way.
To remove + 12, we take away 12.
We do it on both sides, so they stay equal.
+ 12 and − 12 make zero.
Cross them out.
14 minus 12 is 2.
So x = 2.
So x = 2 and y = 4, both at the same time.
We write it as (2, 4).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 2 and y = 4 into ①.
We get 14, the same as the right side.
Now ②.
We get 6, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: make the x numbers opposite.
Then: add the equations.
Then: find y.
Then: find x.
Last: check.