Step by step
The question
Question: solve x + y − z = 3, x − y − z = 1 and x + y + 3z = −1.
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 y from ① and ②
Look at y in ① and ②: y and −y.
They are opposites, so adding removes y.
We write ① and ② under each other: x under x, y under y, z under z.
y and −y make zero.
Cross them out.
We add: the x parts, the z parts, and the numbers.
x + x = 2x.
−z − z = −2z.
3 + 1 = 4.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 2x − 2z = 4.
We call it ④.
It has no y.
2 Remove y from ② and ③
Look at y in ② and ③: −y and y.
They are opposites, so adding removes y.
We write ② and ③ under each other: x under x, y under y, z under z.
−y and y make zero.
Cross them out.
We add: the x parts, the z parts, and the numbers.
x + x = 2x.
−z + 3z = 2z.
1 + (−1) = 0.
Now we have 2x + 2z = 0.
We call it ⑤.
It has no y.
Now ④ and ⑤ are two equations with two letters: x and z.
We solve them like before, by elimination.
We write ④ and ⑤ again, in columns.
3 Add the equations
Look at z: −2z and 2z.
They are opposites.
Added, they make zero.
−2z and 2z make zero.
Cross them out.
We add ④ and ⑤: the x parts together, and the numbers together.
2x + 2x = 4x.
4 + 0 = 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.
On the right too: 4 divided by 4 is 1.
So x = 1.
5 Find z
We put x = 1 into ④.
2 times 1 is 2.
Now we have 2 − 2z = 4.
We want z by itself.
The 2 is in the way.
To remove 2, we take away 2.
We do it on both sides, so they stay equal.
2 and − 2 make zero.
Cross them out.
4 minus 2 is 2.
Now we have −2z = 2.
−2z means −2 times z.
We want just z.
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 z is left.
2 divided by −2 is −1.
So z = −1.
6 Find y
We put x = 1 and z = −1 into ①, the easiest one.
1 − (−1) is 2.
We want y by itself.
The 2 is in the way.
To remove 2, we take away 2.
We do it on both sides, so they stay equal.
2 and − 2 make zero.
Cross them out.
3 minus 2 is 1.
So y = 1.
So x = 1, y = 1 and z = −1, all at the same time.
We write it as (1, 1, −1).
7 Check
Let's check in every equation.
We put x = 1, y = 1, z = −1 into ①.
We get 3, the same as the right side.
Now ②.
We get 1, the same as the right side.
Now ③.
We get −1, the same as the right side.
All of them work.
So the answer is right!
Let's look back.
First: remove y from ① and ②.
Then: remove y from ② and ③.
Then: add the equations.
Then: find x.
Then: find z.
Then: find y.
Last: check.