Step by step
The question
Question: solve x + y + z = 6, 2x + 3y − z = 5 and 3x − 2y + z = 19.
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.
y + 3y = 4y.
6 + 5 = 11.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 3x + 4y = 11.
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 + 3x = 5x.
3y − 2y = y.
5 + 19 = 24.
Now we have 5x + y = 24.
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 y numbers opposite
We want to remove y.
In ④ it is 4y, in ⑤ it is y.
We make them opposite: 4y and −4y.
Then adding removes y.
We multiply every part of ⑤ by −4.
We do it on both sides, so they stay equal.
−4 times 5x is −20x.
−4 times y is −4y.
−4 times 24 is −96.
We call it ⑥.
4 Add the equations
We write ④ and ⑥ under each other: x under x, y under y.
Look at y: 4y and −4y.
They are opposites.
Added, they make zero.
4y and −4y make zero.
Cross them out.
We add ④ and ⑥: the x parts together, and the numbers together.
3x − 20x = −17x.
11 + (−96) = −85.
Now we have −17x = −85.
5 Find x
−17x means −17 times x.
We want just x.
To remove times −17, we divide by −17.
We do it on both sides, so they stay equal.
−17 divided by −17 is 1.
So just x is left.
−85 divided by −17 is 5.
So x = 5.
6 Find y
We put x = 5 into ⑤.
5 times 5 is 25.
Now we have 25 + y = 24.
We want y by itself.
The 25 is in the way.
To remove 25, we take away 25.
We do it on both sides, so they stay equal.
25 and − 25 make zero.
Cross them out.
24 minus 25 is −1.
So y = −1.
7 Find z
We put x = 5 and y = −1 into ①, the easiest one.
5 + (−1) 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.
6 minus 4 is 2.
So z = 2.
So x = 5, y = −1 and z = 2, all at the same time.
We write it as (5, −1, 2).
8 Check
Let's check in every equation.
We put x = 5, y = −1, z = 2 into ①.
We get 6, the same as the right side.
Now ②.
We get 5, the same as the right side.
Now ③.
We get 19, 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 y numbers opposite.
Then: add the equations.
Then: find x.
Then: find y.
Then: find z.
Last: check.