Solve y = 2x + 1, 3x + y = 11 Easy

The question

Question: solve y = 2x + 1 and 3x + y = 11.

We call them ① and ②.

We want the x and y that make both true.

1 Put ① into ②

In ①, y is already alone.

y is the same as 2x + 1.

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

The brackets keep it together.

Putting one thing in place of another is called substitution.

2 Find x

A bracket with a plus or nothing in front can just be removed.

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

3x + 2x = 5x.

Now we have 5x + 1 = 11.

We want x by itself.

The + 1 is in the way.

To remove + 1, we take away 1.

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

+ 1 and − 1 make zero.

Cross them out.

11 minus 1 is 10.

Now we have 5x = 10.

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.

3 Find y

Now we put x = 2 into ①.

That is 4 + 1.

So y = 5.

So x = 2 and y = 5, both at the same time.

We write it as (2, 5).

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

4 Check

Let's check in both equations.

We put x = 2 and y = 5 into ①.

And the right side.

Both sides are 5.

Now ②.

We get 11, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: put ① into ②.

Then: find x.

Then: find y.

Last: check.

▶ Watch it solved

Solve x + 3y = 14; x + y = 6 Medium

The question

Question: solve x + 3y = 14 and x + y = 6.

We call them ① and ②.

We want the x and y that make both true.

We will add the equations to remove one letter.

This way is called elimination.

1 Make the x numbers opposite

We want to remove x.

In ① it is x, in ② it is x.

We make them opposite: x and −x.

Then adding removes x.

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 y is −y.

−1 times 6 is −6.

We call it ③.

2 Add the equations

We write ① and ③ under each other: x under x, y under y.

Look at x: x and −x.

They are opposites.

Added, they make zero.

Why can we add equations?

Both sides of ③ are the same amount.

So we add the same amount to both sides of ①.

x and −x make zero.

Cross them out.

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

3y − y = 2y.

14 + (−6) = 8.

Now we have 2y = 8.

3 Find y

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.

8 divided by 2 is 4.

So y = 4.

4 Find x

We put y = 4 into ①.

3 times 4 is 12.

Now we have x + 12 = 14.

We want x by itself.

The + 12 is in the way.

To remove + 12, we take away 12.

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

+ 12 and − 12 make zero.

Cross them out.

14 minus 12 is 2.

So x = 2.

So x = 2 and y = 4, both at the same time.

We write it as (2, 4).

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

5 Check

Let's check in both equations.

We put x = 2 and y = 4 into ①.

We get 14, the same as the right side.

Now ②.

We get 6, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: make the x numbers opposite.

Then: add the equations.

Then: find y.

Then: find x.

Last: check.

▶ Watch it solved

Solve 3x + 2y = 16, 2x + 3y = 14 Harder

The question

Question: solve 3x + 2y = 16 and 2x + 3y = 14.

We call them ① and ②.

We want the x and y that make both true.

We will add the equations to remove one letter.

This way is called elimination.

1 Make the x numbers opposite

We want to remove x.

In ① it is 3x, in ② it is 2x.

We make them opposite: 6x and −6x.

Then adding removes x.

We multiply every part of ① by 2.

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

2 times 3x is 6x.

2 times 2y is 4y.

2 times 16 is 32.

We call it ③.

We multiply every part of ② by −3.

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

−3 times 2x is −6x.

−3 times 3y is −9y.

−3 times 14 is −42.

We call it ④.

2 Add the equations

We write ③ and ④ under each other: x under x, y under y.

Look at x: 6x and −6x.

They are opposites.

Added, they make zero.

Why can we add equations?

Both sides of ④ are the same amount.

So we add the same amount to both sides of ③.

6x and −6x make zero.

Cross them out.

We add ③ and ④: the y parts together, and the numbers together.

4y − 9y = −5y.

32 + (−42) = −10.

Now we have −5y = −10.

3 Find y

−5y means −5 times y.

We want just y.

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 y is left.

−10 divided by −5 is 2.

So y = 2.

4 Find x

We put y = 2 into ①.

2 times 2 is 4.

Now we have 3x + 4 = 16.

We want x 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.

16 minus 4 is 12.

Now we have 3x = 12.

3x means 3 times x.

We want just x.

To remove times 3, we divide by 3.

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

3 divided by 3 is 1.

So just x is left.

12 divided by 3 is 4.

So x = 4.

So x = 4 and y = 2, both at the same time.

We write it as (4, 2).

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

5 Check

Let's check in both equations.

We put x = 4 and y = 2 into ①.

We get 16, the same as the right side.

Now ②.

We get 14, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: make the x numbers opposite.

Then: add the equations.

Then: find y.

Then: find x.

Last: check.

▶ Watch it solved