Solve x + 3y + z = 10, 2x - y + z = 2, x + y - z = -2

Step by step

The question

Question: solve x + 3y + z = 10, 2x − y + z = 2 and x + y − z = −2.

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.

3y + y = 4y.

10 + (−2) = 8.

Careful!

We add every part of both equations: each letter, and the numbers on the right too.

Now we have 2x + 4y = 8.

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 + x = 3x.

−y + y = 0.

2 + (−2) = 0.

Now we have 3x = 0.

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 3x.

We make them opposite: 6x and −6x.

Then adding removes x.

We multiply every part of ④ by 3.

We do it on both sides, so they stay equal.

3 times 2x is 6x.

3 times 4y is 12y.

3 times 8 is 24.

We call it ⑥.

We multiply every part of ⑤ by −2.

We do it on both sides, so they stay equal.

−2 times 3x is −6x.

−2 times 0 is 0.

We call it ⑦.

4 Add the equations

We write ⑥ and ⑦ under each other: x under x, y under y.

Look at x: 6x and −6x.

They are opposites.

Added, they make zero.

6x and −6x make zero.

Cross them out.

We add ⑥ and ⑦: the y parts together, and the numbers together.

12y + 0y = 12y.

24 + 0 = 24.

Now we have 12y = 24.

5 Find y

12y means 12 times y.

We want just y.

To remove times 12, we divide by 12.

We do it on both sides, so they stay equal.

12 divided by 12 is 1.

So just y is left.

24 divided by 12 is 2.

So y = 2.

6 Find x

We put y = 2 into ④.

4 times 2 is 8.

Now we have 2x + 8 = 8.

We want x 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.

8 minus 8 is 0.

Now we have 2x = 0.

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.

0 divided by 2 is 0.

So x = 0.

7 Find z

We put x = 0 and y = 2 into ③, the easiest one.

0 + 2 is 2.

We want z 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.

−2 minus 2 is −4.

Now we have −z = −4.

−z means −1 times z.

We want just z.

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 z is left.

−4 divided by −1 is 4.

So z = 4.

So x = 0, y = 2 and z = 4, all at the same time.

We write it as (0, 2, 4).

8 Check

Let's check in every equation.

We put x = 0, y = 2, z = 4 into ①.

We get 10, the same as the right side.

Now ②.

We get 2, the same as the right side.

Now ③.

We get −2, 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.