Step by step
The question
Question: solve x + 3y + z = 12, x − y − z = 0 and x − y + z = 8.
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.
3y − y = 2y.
12 + 0 = 12.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 2x + 2y = 12.
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.
0 + 8 = 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.
2x + 2x = 4x.
12 + 8 = 20.
Now we have 4x = 20.
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.
20 divided by 4 is 5.
So x = 5.
5 Find y
We put x = 5 into ④.
2 times 5 is 10.
Now we have 10 + 2y = 12.
We want y by itself.
The 10 is in the way.
To remove 10, we take away 10.
We do it on both sides, so they stay equal.
10 and − 10 make zero.
Cross them out.
12 minus 10 is 2.
Now we have 2y = 2.
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.
On the right too: 2 divided by 2 is 1.
So y = 1.
6 Find z
We put x = 5 and y = 1 into ②, the easiest one.
5 − 1 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.
0 minus 4 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 = 5, y = 1 and z = 4, all at the same time.
We write it as (5, 1, 4).
7 Check
Let's check in every equation.
We put x = 5, y = 1, z = 4 into ①.
We get 12, the same as the right side.
Now ②.
We get 0, the same as the right side.
Now ③.
We get 8, 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.