Step by step
The question
Question: solve x − 2y + z = −3, 2x + y − z = 13 and x + y + z = 3.
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 + 2x = 3x.
−2y + y = −y.
−3 + 13 = 10.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 3x − y = 10.
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.
2x + x = 3x.
y + y = 2y.
13 + 3 = 16.
Now we have 3x + 2y = 16.
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 Make the x numbers opposite
We want to remove x.
In ④ it is 3x, in ⑤ it is 3x.
We make them opposite: 3x and −3x.
Then adding removes x.
We multiply every part of ⑤ by −1.
We do it on both sides, so they stay equal.
−1 times 3x is −3x.
−1 times 2y is −2y.
−1 times 16 is −16.
We call it ⑥.
4 Add the equations
We write ④ and ⑥ under each other: x under x, y under y.
Look at x: 3x and −3x.
They are opposites.
Added, they make zero.
3x and −3x make zero.
Cross them out.
We add ④ and ⑥: the y parts together, and the numbers together.
−y − 2y = −3y.
10 + (−16) = −6.
Now we have −3y = −6.
5 Find y
−3y means −3 times y.
We want just y.
To remove times −3, we divide by −3.
We do it on both sides, so they stay equal.
−3 divided by −3 is 1.
So just y is left.
−6 divided by −3 is 2.
So y = 2.
6 Find x
We put y = 2 into ④.
−1 times 2 is −2.
Now we have 3x − 2 = 10.
We want x 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 12.
Now we have 3x = 12.
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
12 divided by 3 is 4.
So x = 4.
7 Find z
We put x = 4 and y = 2 into ③, the easiest one.
4 + 2 is 6.
We want z 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.
3 minus 6 is −3.
So z = −3.
So x = 4, y = 2 and z = −3, all at the same time.
We write it as (4, 2, −3).
8 Check
Let's check in every equation.
We put x = 4, y = 2, z = −3 into ①.
We get −3, the same as the right side.
Now ②.
We get 13, the same as the right side.
Now ③.
We get 3, 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: make the x numbers opposite.
Then: add the equations.
Then: find y.
Then: find x.
Then: find z.
Last: check.