Step by step
The question
Question: solve 2x − y + z = 3, x + 2y − z = 8 and x + y + 2z = 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.
2x + x = 3x.
−y + 2y = y.
3 + 8 = 11.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 3x + y = 11.
We call it ④.
It has no z.
2 Remove z from ① and ③
Look at z in ① and ③: z and 2z.
We make them opposite: −2z and 2z.
Then adding removes z.
We multiply every part of ① by −2.
We do it on both sides, so they stay equal.
−2 times 2x is −4x.
−2 times −y is 2y.
−2 times z is −2z.
−2 times 3 is −6.
We write ① times −2 and ③ under each other: x under x, y under y, z under z.
−2z and 2z make zero.
Cross them out.
We add: the x parts, the y parts, and the numbers.
−4x + x = −3x.
2y + y = 3y.
−6 + 3 = −3.
Now we have −3x + 3y = −3.
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 x: 3x and −3x.
They are opposites.
Added, they make zero.
3x and −3x make zero.
Cross them out.
We add ④ and ⑤: the y parts together, and the numbers together.
y + 3y = 4y.
11 + (−3) = 8.
Now we have 4y = 8.
4 Find y
4y means 4 times y.
We want just y.
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 y is left.
8 divided by 4 is 2.
So y = 2.
5 Find x
We put y = 2 into ④.
We want x 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.
11 minus 2 is 9.
Now we have 3x = 9.
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.
9 divided by 3 is 3.
So x = 3.
6 Find z
We put x = 3 and y = 2 into ①, the easiest one.
2(3) − 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.
So z = −1.
So x = 3, y = 2 and z = −1, all at the same time.
We write it as (3, 2, −1).
7 Check
Let's check in every equation.
We put x = 3, y = 2, z = −1 into ①.
We get 3, the same as the right side.
Now ②.
We get 8, 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 ① and ③.
Then: add the equations.
Then: find y.
Then: find x.
Then: find z.
Last: check.