Question

How do you use the point on the line and the slope of the line to find three additional points through which the line passes: Point: (7, -2) Slope:m = 1/2?

Answer 1

Here's how you can do that.

Explanation:

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 `Deltay`, divided by the change in `x`, or `Deltax`

`m = (Deltay)/(Deltax)`

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 implies {(Deltay = 1), (Deltax = 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) :} implies (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) :} implies (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 = 1/2x - 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 line.