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