Step by step
The question
Question: solve x − y − z = 0, x + y − z = 10 and x + y + 3z = 6.
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 ②
Look at y in ① and ②: −y and y.
They are opposites, so adding removes y.
We write ① and ② 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.
−z − z = −2z.
0 + 10 = 10.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 2x − 2z = 10.
We call it ④.
It has no y.
2 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.
x + x = 2x.
−z + 3z = 2z.
0 + 6 = 6.
Now we have 2x + 2z = 6.
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.
2x + 2x = 4x.
10 + 6 = 16.
Now we have 4x = 16.
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.
16 divided by 4 is 4.
So x = 4.
5 Find z
We put x = 4 into ④.
2 times 4 is 8.
Now we have 8 − 2z = 10.
We want z by itself.
The 8 is in the way.
To remove 8, we take away 8.
We do it on both sides, so they stay equal.
8 and − 8 make zero.
Cross them out.
10 minus 8 is 2.
Now we have −2z = 2.
−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.
2 divided by −2 is −1.
So z = −1.
6 Find y
We put x = 4 and z = −1 into ①, the easiest one.
4 − (−1) is 5.
We want y by itself.
The 5 is in the way.
To remove 5, we take away 5.
We do it on both sides, so they stay equal.
5 and − 5 make zero.
Cross them out.
0 minus 5 is −5.
Now we have −y = −5.
−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.
−5 divided by −1 is 5.
So y = 5.
So x = 4, y = 5 and z = −1, all at the same time.
We write it as (4, 5, −1).
7 Check
Let's check in every equation.
We put x = 4, y = 5, z = −1 into ①.
We get 0, the same as the right side.
Now ②.
We get 10, the same as the right side.
Now ③.
We get 6, 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 ②.
Then: remove y from ① and ③.
Then: add the equations.
Then: find x.
Then: find z.
Then: find y.
Last: check.