Step by step
The question
Question: solve x + y + z = 7, x − y + z = 3 and x + y − z = 5.
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 ③
Look at z in ① and ③: z and −z.
They are opposites, so adding removes z.
We write ① 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.
7 + 5 = 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 = 0.
3 + 5 = 8.
Now we have 2x = 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 Make the x numbers opposite
We want to remove x.
In ④ it is 2x, in ⑤ it is 2x.
We make them opposite: 2x and −2x.
Then adding removes x.
We multiply every part of ⑤ by −1.
We do it on both sides, so they stay equal.
−1 times 2x is −2x.
−1 times 8 is −8.
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: 2x and −2x.
They are opposites.
Added, they make zero.
2x and −2x make zero.
Cross them out.
We add ④ and ⑥: the y parts together, and the numbers together.
2y + 0y = 2y.
12 + (−8) = 4.
Now we have 2y = 4.
5 Find y
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.
4 divided by 2 is 2.
So y = 2.
6 Find x
We put y = 2 into ④.
2 times 2 is 4.
Now we have 2x + 4 = 12.
We want x 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.
12 minus 4 is 8.
Now we have 2x = 8.
2x means 2 times x.
We want just x.
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 x is left.
8 divided by 2 is 4.
So x = 4.
7 Find z
We put x = 4 and y = 2 into ①, the easiest one.
4 + 2 is 6.
We want z by itself.
The 6 is in the way.
To remove 6, we take away 6.
We do it on both sides, so they stay equal.
6 and − 6 make zero.
Cross them out.
7 minus 6 is 1.
So z = 1.
So x = 4, y = 2 and z = 1, all at the same time.
We write it as (4, 2, 1).
8 Check
Let's check in every equation.
We put x = 4, y = 2, z = 1 into ①.
We get 7, the same as the right side.
Now ②.
We get 3, the same as the right side.
Now ③.
We get 5, 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 ③.
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.