Step by step
The question
Question: solve x − y − z = −9, x − y + z = −1 and x + 3y + z = 15.
We call them ①, ② and ③.
We want the x, y and z that make all three true.
The plan: remove one letter twice.
Then we have two equations with two letters, and we know how to solve those.
1 Remove z from ① and ②
Look at z in ① and ②: −z and z.
They are opposites, so adding removes z.
We write ① and ② under each other: x under x, y under y, z under z.
−z and z make zero.
Cross them out.
We add: the x parts, the y parts, and the numbers.
x + x = 2x.
−y − y = −2y.
−9 + (−1) = −10.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 2x − 2y = −10.
We call it ④.
It has no z.
2 Remove z from ① and ③
Look at z in ① and ③: −z and z.
They are opposites, so adding removes z.
We write ① and ③ under each other: x under x, y under y, z under z.
−z and z make zero.
Cross them out.
We add: the x parts, the y parts, and the numbers.
x + x = 2x.
−y + 3y = 2y.
−9 + 15 = 6.
Now we have 2x + 2y = 6.
We call it ⑤.
It has no z.
Now ④ and ⑤ are two equations with two letters: x and y.
We solve them like before, by elimination.
We write ④ and ⑤ again, in columns.
3 Add the equations
Look at y: −2y and 2y.
They are opposites.
Added, they make zero.
−2y and 2y make zero.
Cross them out.
We add ④ and ⑤: the x parts together, and the numbers together.
2x + 2x = 4x.
−10 + 6 = −4.
Now we have 4x = −4.
4 Find x
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.
−4 divided by 4 is −1.
So x = −1.
5 Find y
We put x = −1 into ④.
2 times −1 is −2.
Now we have −2 − 2y = −10.
We want y by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
−2 and + 2 make zero.
Cross them out.
−10 plus 2 is −8.
Now we have −2y = −8.
−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.
6 Find z
We put x = −1 and y = 4 into ①, the easiest one.
−1 − 4 is −5.
We want z by itself.
The −5 is in the way.
To remove −5, we add 5.
We do it on both sides, so they stay equal.
−5 and + 5 make zero.
Cross them out.
−9 plus 5 is −4.
Now we have −z = −4.
−z means −1 times z.
We want just z.
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 z is left.
−4 divided by −1 is 4.
So z = 4.
So x = −1, y = 4 and z = 4, all at the same time.
We write it as (−1, 4, 4).
7 Check
Let's check in every equation.
We put x = −1, y = 4, z = 4 into ①.
We get −9, the same as the right side.
Now ②.
We get −1, the same as the right side.
Now ③.
We get 15, the same as the right side.
All of them work.
So the answer is right!
Let's look back.
First: remove z from ① and ②.
Then: remove z from ① and ③.
Then: add the equations.
Then: find x.
Then: find y.
Then: find z.
Last: check.