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?

1 Answer
Jan 9, 2017

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.