Step by step
The question
Question: solve 2x + 3y + z = −1, x − y − z = 3 and x − y + z = 5.
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.
2x + x = 3x.
3y − y = 2y.
−1 + 3 = 2.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 3x + 2y = 2.
We call it ④.
It has no z.
2 Remove z from ② and ③
Look at z in ② and ③: −z and z.
They are opposites, so adding removes z.
We write ② 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 − y = −2y.
3 + 5 = 8.
Now we have 2x − 2y = 8.
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.
3x + 2x = 5x.
2 + 8 = 10.
Now we have 5x = 10.
4 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.
10 divided by 5 is 2.
So x = 2.
5 Find y
We put x = 2 into ④.
3 times 2 is 6.
Now we have 6 + 2y = 2.
We want y by itself.
The 6 is in the way.
To remove 6, we take away 6.
We do it on both sides, so they stay equal.
6 and − 6 make zero.
Cross them out.
2 minus 6 is −4.
Now we have 2y = −4.
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.
−4 divided by 2 is −2.
So y = −2.
6 Find z
We put x = 2 and y = −2 into ②, the easiest one.
2 − (−2) is 4.
We want z by itself.
The 4 is in the way.
To remove 4, we take away 4.
We do it on both sides, so they stay equal.
4 and − 4 make zero.
Cross them out.
3 minus 4 is −1.
Now we have −z = −1.
−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.
−1 divided by −1 is 1.
So z = 1.
So x = 2, y = −2 and z = 1, all at the same time.
We write it as (2, −2, 1).
7 Check
Let's check in every equation.
We put x = 2, y = −2, z = 1 into ①.
We get −1, the same as the right side.
Now ②.
We get 3, the same as the right side.
Now ③.
We get 5, 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 ② and ③.
Then: add the equations.
Then: find x.
Then: find y.
Then: find z.
Last: check.