Step by step
The question
Question: solve x + y + z = 2, x − y + z = 4 and 3x + y + 3z = 10.
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.
2 + 4 = 6.
Careful!
We add every part of both equations: each letter, and the numbers on the right too.
Now we have 2x + 2z = 6.
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 + 3x = 4x.
z + 3z = 4z.
4 + 10 = 14.
Now we have 4x + 4z = 14.
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 Make the x numbers opposite
We want to remove x.
In ④ it is 2x, in ⑤ it is 4x.
We make them opposite: −4x and 4x.
Then adding removes x.
We multiply every part of ④ by −2.
We do it on both sides, so they stay equal.
−2 times 2x is −4x.
−2 times 2z is −4z.
−2 times 6 is −12.
We call it ⑥.
4 Add the equations
We write ⑥ and ⑤ under each other: x under x, z under z.
Look at x: −4x and 4x.
They are opposites.
Added, they make zero.
−4x and 4x make zero.
Cross them out.
We add ⑥ and ⑤: the z parts together, and the numbers together.
−4z + 4z = 0.
−12 + 14 = 2.
We get 0 = 2.
Both letters are gone, and 0 = 2 is never true.
So the three equations can never all be true together: there is no solution.
Let's look back.
First: remove y from ① and ②.
Then: remove y from ② and ③.
Then: make the x numbers opposite.
Last: add the equations.