Solve the simultaneous equations x + 2y = 8 and x − y = 2. Show your…

Step by step: way 1, elimination

The question

Question: solve x + 2y = 8 and x − y = 2.

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 x parts opposite

We want to remove one letter.

In ① it is x, in ② it is x.

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

We make them opposite: x and −x.

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

Then adding removes x.

x times −1 is −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 2 is −2.

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.

x and −x make zero.

Cross them out.

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

2y + y = 3y.

8 + (−2) = 6.

Now we have 3y = 6.

Is y alone now?

Not yet: 3y means 3 times y.

How do we remove it?

3 Find y

To remove times 3, we divide by 3.

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

3 divided by 3 is 1.

They cancel each other.

So just y is left.

6 divided by 3 is 2.

So y = 2.

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

4 Find x

We put y = 2 into ①.

2 times 2 is 4.

Now we have x + 4 = 8.

We want x by itself.

The + 4 is in the way.

Is x alone now?

Not yet: + 4 is still with x.

To remove + 4, we take away 4.

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

+ 4 and − 4 cancel each other.

They make zero.

8 minus 4 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 8, the same as the right side.

Now ②.

We get 2, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: make the x parts opposite.

Then: add the equations.

Then: find y.

Then: find x.

Last: check.

Step by step: way 2, substitution

The question

Question: solve x + 2y = 8 and x − y = 2 by substitution.

We call them ① and ②.

We want the x and y that make both true.

1 Get x alone in ①

No letter is alone yet.

In ①, 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 + 8③.

2 Put ③ into ②

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

In ③, x is now alone.

x is the same as −2y + 8.

So in ②, we write (−2y + 8) in place of x.

The brackets keep it together.

Putting one thing in place of another is called substitution.

3 Find y

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

All the y terms go together.

−2y − y = −3y.

Now we have −3y + 8 = 2.

We want y by itself.

The + 8 is in the way.

Is y alone now?

Not yet: + 8 is still with y.

How do we remove it?

To remove + 8, we take away 8.

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

+ 8 and − 8 cancel each other.

They make zero.

2 minus 8 is −6.

Now we have −3y = −6.

−3y means −3 times y.

Is y alone now?

Not yet: −3y means −3 times y.

To remove times −3, we divide by −3.

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

−3 divided by −3 is 1.

They cancel each other.

So just y is left.

−6 divided by −3 is 2.

So y = 2.

So after step 3: we took away 8 and divided by 3, and now we have y = 2.

4 Find x

Now we put y = 2 into ③.

Times first.

That is −4 + 8.

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 8, the same as the right side.

Now ②.

We get 2, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: get x alone in ①.

Then: put ③ into ②.

Then: find y.

Then: find x.

Last: check.

Step by step: way 3, graphing

The question

Question: solve by graphing x + 2y = 8 and x − y = 2.

The answer is a point that is on both lines.

So we draw both lines and see where they meet.

1 Crossing points of ①

Line ① is not y = mx + b.

So we find where it crosses the two axes.

Two points are enough for a line.

On the x-axis, y = 0.

Put 0 in place of y.

2 times 0 is 0.

Adding 0 changes nothing.

So x = 8.

On the y-axis, x = 0.

Put 0 in place of x.

0 plus 2y is just 2y.

2y means 2 times y.

Is y alone now?

Not yet: 2y means 2 times y.

How do we remove it?

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.

8 divided by 2 is 4.

So y = 4.

So line ① crosses the axes at (8, 0) and (0, 4).

So after step 1: we divided by 2, and now we have y = 4.

2 Crossing points of ②

Line ② is not y = mx + b.

So we find where it crosses the two axes.

Two points are enough for a line.

On the x-axis, y = 0.

Put 0 in place of y.

−y means minus 1 times y.

1 times 0 is 0.

Taking away 0 changes nothing.

So x = 2.

On the y-axis, x = 0.

Put 0 in place of x.

0 minus y is −y.

−y means −1 times y.

Is y alone now?

Not yet: −y means −1 times y.

To remove times −1, we divide by −1.

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

−1 divided by −1 is 1.

They cancel each other.

So just y is left.

Plus over minus is minus.

2 divided by −1 is −2.

So y = −2.

So line ② crosses the axes at (2, 0) and (0, −2).

So after step 2: we divided by 1, and now we have y = −2.

3 Graph line ①

Draw the grid: the x-axis goes across, and the y-axis goes up and down.

They meet at 0, a point called the origin.

Put the two crossing points on the grid,

and join them.

4 Graph line ②

Put the two crossing points on the grid,

and join them.

5 Where they meet

The lines meet at one point.

That point is on both lines, so it makes both equations true.

Go straight down to the x-axis: x = 4.

Go straight across to the y-axis: y = 2.

So x = 4 and y = 2, both at the same time: the point (4, 2).

6 Check in both

Let's check: put x = 4 and y = 2 into both equations.

First ①.

We get 8, the same as the right side.

It works.

Now ②.

We get 2, the same as the right side.

It works.

(4, 2) is on both lines.

So the answer is right!

Let's look back.

First: crossing points of ①.

Then: crossing points of ②.

Then: graph line ①.

Then: graph line ②.

Then: where they meet.

Last: check in both.