Step by step
The question
Question: solve x + y = 5 and x + y = 8.
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 8 is −8.
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.
y − y = 0.
5 + (−8) = −3.
We get 0 = −3.
Both letters are gone, and 0 = −3 is never true.
The two lines never meet.
So there is no solution.
Let's look back.
First: make the x numbers opposite.
Last: add the equations.