Find the coordinates of three more points that lie on the line passing through the points (0, 0) and (3, -4). Explain your method.

All you need to know here is that the slope of the line contains a set of directions that allow you to start from a point that lies on a given line and find other points that lie on the same line.

So, you know that a given line has a slope of

m

=

1

2

As you know, the slope of a line is defined as the change in  

y

, or  

Δ

y

, divided by the change in  

x

, or  

Δ

x

m

=

Δ

y

Δ

x

Now, you know that the point  

(

7

,

−

2

)

lies on this line. The change in  

y

tells you the number of positions that you must move up on the  

y

axis in order to find the  

y

-coordinate of another point that lies on the line.

Similarly, the change in  

x

tells you the number of positions that you must move to the right on the  

x

axis in order to find the  

x

coordinate of another point that lies on the line.

In this case, you have

m

=

1

2

⇒

{

Δ

y

=

1

Δ

x

=

2

So, if you start at  

x

=

7

, you must move  

2

positions to the right to find

x

2

=

7

+

2

=

9

Similarly, if you start at  

y

=

−

2

, you mus move  

1

position up to find

y

2

=

−

2

+

1

=

−

1

Therefore, a second point on the given line is  

(

9

,

−

1

)

.

Now here comes the cool part, You can use multiples of the slope to find additional points by starting from the same point  

(

7

,

−

2

)

. For example, you have

m

=

1

2

=

2

4

This means that you will get

{

x

3

=

7

+

4

=

11

y

3

=

−

2

+

2

=

0

⇒

(

11

,

0

)

is another point that lies on the line

Similarly, you can also have

m

=

1

2

=

−

1

−

2

In this case, you're moving  

2

positions to the left for  

x

and  

1

position down for  

y

.

This means that

{

x

4

=

7

+

(

−

2

)

=

5

y

4

=

−

2

+

(

−

1

)

=

−

3

⇒

(

5

,

−

3

)

is another point that lies on the line

Therefore, you can say that  

(

5

,

−

3

)

,  

(

7

,

−

2

)

,  

(

9

,

−

1

)

, and  

(

11

,

0

)

are all points that lie on the given line.

To double-check the result, use one of the points to write the equation of the line

(

y

−

y

4

)

=

m

⋅

(

x

−

x

4

)

y

−

0

=

1

2

⋅

(

x

−

11

)

y

=

1

2

x

−

11

2

The line looks like this

graph{1/2x - 11/2 [-10, 10, -5, 5]}

As you can see, all the points that we've found lie on the li

Step-by-step explanation: