Step by step
The question
Question: solve 2x + y − z = −3, x + y + 3z = 18 and x − y − z = −10.
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 ③
Look at y in ① and ③: y and −y.
They are opposites, so adding removes y.
We write ① 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.
2x + x = 3x.
−z − z = −2z.
−3 + (−10) = −13.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 3x − 2z = −13.
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.
3z − z = 2z.
18 + (−10) = 8.
Now we have 2x + 2z = 8.
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.
3x + 2x = 5x.
−13 + 8 = −5.
Now we have 5x = −5.
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.
−5 divided by 5 is −1.
So x = −1.
5 Find z
We put x = −1 into ④.
3 times −1 is −3.
Now we have −3 − 2z = −13.
We want z by itself.
The − 3 is in the way.
To remove − 3, we add 3.
We do it on both sides, so they stay equal.
−3 and + 3 make zero.
Cross them out.
−13 plus 3 is −10.
Now we have −2z = −10.
−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.
−10 divided by −2 is 5.
So z = 5.
6 Find y
We put x = −1 and z = 5 into ③, the easiest one.
−1 − 5 is −6.
We want y 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.
−10 plus 6 is −4.
Now we have −y = −4.
−y means −1 times y.
We want just y.
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 y is left.
−4 divided by −1 is 4.
So y = 4.
So x = −1, y = 4 and z = 5, all at the same time.
We write it as (−1, 4, 5).
7 Check
Let's check in every equation.
We put x = −1, y = 4, z = 5 into ①.
We get −3, the same as the right side.
Now ②.
We get 18, the same as the right side.
Now ③.
We get −10, 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 ③.
Then: remove y from ② and ③.
Then: add the equations.
Then: find x.
Then: find z.
Then: find y.
Last: check.