Step by step
The question
Question: solve 2x + y + 3z = 18, x + y − z = 1 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 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.
2x + x = 3x.
3z − z = 2z.
18 + 3 = 21.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 3x + 2z = 21.
We call it ④.
It has no y.
2 Remove y from ② and ③
Look at y in ② and ③: y and −y.
They are opposites, so adding removes y.
We write ② 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 − z = −2z.
1 + 3 = 4.
Now we have 2x − 2z = 4.
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.
3x + 2x = 5x.
21 + 4 = 25.
Now we have 5x = 25.
4 Find x
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just x is left.
25 divided by 5 is 5.
So x = 5.
5 Find z
We put x = 5 into ④.
3 times 5 is 15.
Now we have 15 + 2z = 21.
We want z by itself.
The 15 is in the way.
To remove 15, we take away 15.
We do it on both sides, so they stay equal.
15 and − 15 make zero.
Cross them out.
21 minus 15 is 6.
Now we have 2z = 6.
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.
6 divided by 2 is 3.
So z = 3.
6 Find y
We put x = 5 and z = 3 into ②, the easiest one.
5 − 3 is 2.
We want y 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.
1 minus 2 is −1.
So y = −1.
So x = 5, y = −1 and z = 3, all at the same time.
We write it as (5, −1, 3).
7 Check
Let's check in every equation.
We put x = 5, y = −1, z = 3 into ①.
We get 18, the same as the right side.
Now ②.
We get 1, 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 y from ① and ③.
Then: remove y from ② and ③.
Then: add the equations.
Then: find x.
Then: find z.
Then: find y.
Last: check.