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Step by step: way 1, elimination

The question

Question: solve 3x + 2y = 380 and x + 2y = 220.

We call them ① and ②.

We want the x and y that make both true.

No letter is alone here.

So we add the equations to remove one letter.

This way is called elimination.

1 Make the y parts opposite

We want to remove one letter.

In ① it is 2y, in ② it is 2y.

2y and 2y are the same size, so y is the easiest letter to remove.

We make them opposite: 2y and −2y.

Opposites are the same size, one plus and one minus.

Then adding removes y.

2y times −1 is −2y.

We multiply every part of ② by −1.

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

−1 times x is −x.

−1 times 2y is −2y.

−1 times 220 is −220.

We call it ③.

2 Add the equations

We write ① and ③ under each other.

x under x, y under y.

Look at y: 2y and −2y.

They are opposites.

Added, they make zero.

2y and −2y make zero.

Cross them out.

We add ① and ③: the x parts together, and the numbers together.

3x − x = 2x.

380 + (−220) = 160.

Now we have 2x = 160.

Is x alone now?

Not yet: 2x means 2 times x.

How do we remove it?

3 Find 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.

They cancel each other.

So just x is left.

160 divided by 2 is 80.

So x = 80.

So after step 3: we divided by 2, and now we have x = 80.

4 Find y

We put x = 80 into ①.

3 times 80 is 240.

Now we have 240 + 2y = 380.

We want y by itself.

The 240 is in the way.

Is y alone now?

Not yet: + 240 is still with y.

To remove 240, we take away 240.

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

240 and − 240 cancel each other.

They make zero.

380 minus 240 is 140.

Now we have 2y = 140.

2y means 2 times y.

Is y alone now?

Not yet: 2y means 2 times 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.

They cancel each other.

So just y is left.

140 divided by 2 is 70.

So y = 70.

So x = 80 and y = 70, both at the same time.

We write it as (80, 70).

On a graph, it is the point where the two lines meet.

5 Check

Let's check in both equations.

We put x = 80 and y = 70 into ①.

We get 380, the same as the right side.

Now ②.

We get 220, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: make the y parts opposite.

Then: add the equations.

Then: find x.

Then: find y.

Last: check.

Step by step: way 2, substitution

The question

Question: solve 3x + 2y = 380 and x + 2y = 220 by substitution.

We call them ① and ②.

We want the x and y that make both true.

1 Get x alone in ②

No letter is alone yet.

In ②, x has no number in front: that means 1 x.

So we get x alone there first.

Then we don't have to divide, so no fractions.

We want x by itself.

The + 2y is in the way.

To remove + 2y, we take away 2y.

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

+ 2y and − 2y cancel each other.

They make zero.

We write it again, without the crossed-out parts.

We write the y part first.

Each part keeps its sign.

x is by itself.

We call it ③.

So after step 1: we took away 2y, and now we have x = −2y + 220③.

2 Put ③ into ①

③ came from ②, so we put it into the other one, ①.

In ③, x is now alone.

x is the same as −2y + 220.

So in ①, we write (−2y + 220) in place of x.

The brackets keep it together.

Putting one thing in place of another is called substitution.

3 Find y

The 3 in front of the bracket multiplies every part inside.

So we write 3 times each part.

Careful!

Not only the first part!

220 is multiplied by 3 too.

Plus times minus is minus.

3 times −2y is −6y.

3 times 220 is 660.

All the y terms go together.

−6y + 2y = −4y.

Now we have −4y + 660 = 380.

We want y by itself.

The + 660 is in the way.

Is y alone now?

Not yet: + 660 is still with y.

How do we remove it?

To remove + 660, we take away 660.

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

+ 660 and − 660 cancel each other.

They make zero.

380 minus 660 is −280.

Now we have −4y = −280.

−4y means −4 times y.

Is y alone now?

Not yet: −4y means −4 times y.

To remove times −4, we divide by −4.

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

−4 divided by −4 is 1.

They cancel each other.

So just y is left.

−280 divided by −4 is 70.

So y = 70.

So after step 3: we took away 660 and divided by 4, and now we have y = 70.

4 Find x

Now we put y = 70 into ③.

Times first.

That is −140 + 220.

So x = 80.

So x = 80 and y = 70, both at the same time.

We write it as (80, 70).

On a graph, it is the point where the two lines meet.

5 Check

Let's check in both equations.

We put x = 80 and y = 70 into ①.

We get 380, the same as the right side.

Now ②.

We get 220, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: get x alone in ②.

Then: put ③ into ①.

Then: find y.

Then: find x.

Last: check.