Step by step
The question
Question: solve 4x + 3y = 2 and 2x − 5y = 14.
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 4x, in ② it is 2x.
We make them opposite: 4x and −4x.
Then adding removes x.
We multiply every part of ② by −2.
We do it on both sides, so they stay equal.
−2 times 2x is −4x.
−2 times −5y is 10y.
−2 times 14 is −28.
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: 4x and −4x.
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.
4x and −4x make zero.
Cross them out.
We add ① and ③: the y parts together, and the numbers together.
3y + 10y = 13y.
2 + (−28) = −26.
Now we have 13y = −26.
3 Find y
13y means 13 times y.
We want just y.
To remove times 13, we divide by 13.
We do it on both sides, so they stay equal.
13 divided by 13 is 1.
So just y is left.
−26 divided by 13 is −2.
So y = −2.
4 Find x
We put y = −2 into ①.
3 times −2 is −6.
Now we have 4x − 6 = 2.
We want x by itself.
The − 6 is in the way.
To remove − 6, we add 6.
We do it on both sides, so they stay equal.
− 6 and + 6 make zero.
Cross them out.
2 plus 6 is 8.
Now we have 4x = 8.
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
8 divided by 4 is 2.
So x = 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.
5 Check
Let's check in both equations.
We put x = 2 and y = −2 into ①.
We get 2, the same as the right side.
Now ②.
We get 14, 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.