Solve x + y - z = 0, x - y + 3z = 8, x - y - z = -4 Easy

The question

Question: solve x + y − z = 0, x − y + 3z = 8 and x − y − z = −4.

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 + 3z = 2z.

0 + 8 = 8.

Careful!

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

Now we have 2x + 2z = 8.

We call it ④.

It has no y.

2 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.

x + x = 2x.

−z − z = −2z.

0 + (−4) = −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.

2x + 2x = 4x.

8 + (−4) = 4.

Now we have 4x = 4.

4 Find x

4x means 4 times x.

We want just x.

To remove times 4, we divide by 4.

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

4 divided by 4 is 1.

So just x is left.

On the right too: 4 divided by 4 is 1.

So x = 1.

5 Find z

We put x = 1 into ④.

2 times 1 is 2.

Now we have 2 + 2z = 8.

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.

8 minus 2 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 = 1 and z = 3 into ①, the easiest one.

1 − 3 is −2.

We want y by itself.

The −2 is in the way.

To remove −2, we add 2.

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

−2 and + 2 make zero.

Cross them out.

0 plus 2 is 2.

So y = 2.

So x = 1, y = 2 and z = 3, all at the same time.

We write it as (1, 2, 3).

7 Check

Let's check in every equation.

We put x = 1, y = 2, z = 3 into ①.

We get 0, the same as the right side.

Now ②.

We get 8, the same as the right side.

Now ③.

We get −4, 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 ②.

Then: remove y from ① and ③.

Then: add the equations.

Then: find x.

Then: find z.

Then: find y.

Last: check.

▶ Watch it solved

Solve 2x + 3y + z = -1, x - y - z = 3, x - y + z = 5 Medium

The question

Question: solve 2x + 3y + z = −1, 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 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.

2x + x = 3x.

3y − y = 2y.

−1 + 3 = 2.

Careful!

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

Now we have 3x + 2y = 2.

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 = −2y.

3 + 5 = 8.

Now we have 2x − 2y = 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 Add the equations

Look at y: 2y and −2y.

They are opposites.

Added, they make zero.

2y and −2y make zero.

Cross them out.

We add ④ and ⑤: the x parts together, and the numbers together.

3x + 2x = 5x.

2 + 8 = 10.

Now we have 5x = 10.

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.

10 divided by 5 is 2.

So x = 2.

5 Find y

We put x = 2 into ④.

3 times 2 is 6.

Now we have 6 + 2y = 2.

We want y 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.

2 minus 6 is −4.

Now we have 2y = −4.

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 z

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

2 − (−2) 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.

3 minus 4 is −1.

Now we have −z = −1.

−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.

−1 divided by −1 is 1.

So z = 1.

So x = 2, y = −2 and z = 1, all at the same time.

We write it as (2, −2, 1).

7 Check

Let's check in every equation.

We put x = 2, y = −2, z = 1 into ①.

We get −1, 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 ②.

Then: remove z from ② and ③.

Then: add the equations.

Then: find x.

Then: find y.

Then: find z.

Last: check.

▶ Watch it solved

Solve x + y + z = 6, 2x + 3y - z = 5, 3x - 2y + z = 19 Harder

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.

▶ Watch it solved