The Slope of a Line
Find the slope of the line given two points:
Consider two methods as alternatives for finding the slope: (without using the “slope-formulaâ€)
METHOD 1:
First, BUILD a TABLE starting with the given points in the middle of the table and find their
common differences, dx and dy, then we can use these differences to find the x-sequence
and y-sequence.
Since both columns must be arithmetic sequences, to get the next value below for the
x-sequence add dx , then add - dx to get the next value above to complete the x-sequence.
Similarly, to get the next value below for the y-sequence add dy, then add - dy to get the next
value above, and complete the y-sequence. (Always check both sequences.)
Recall that the slope is
and RECORD the value for m.
METHOD 2:
Plot BOTH points on a grid and move up/(down) and over to FIND values for dy and dx
and RECORD the value of m.
Now move using dy and dx to find the next point and continue up/over or down/over
to find several points. Label these points and put them in order in a table of sequences.
Example 1:
1. Given (2, -1) and (- 3, 2) plot the points and draw the line through the points.
Put the points in the middle of the table and find the differences dx and dy.
See work below:
![](./articles_imgs/1055/slope-72.gif)
We found the slope![](./articles_imgs/1055/slope-73.gif)
NOTE: A “neat†method for finding the slope (without using the “formulaâ€) is shown
below:
Ask “how far is it †from y1 to y2 in which direction (dy)
and “how far is it “ from x1 to x2 in which direction (dx).
![](./articles_imgs/1055/slope-74.gif)
Generally, it is better to place the points in the table or chart with x1 < x2
so that dx > 0.
These methods for finding the slope (without using the “formulaâ€) are illustrated in
the example below:
Example 2.
Given points: (3, 2), (5, -3) Find Dx and Dy directly from the table or the points.
Either order gives the same slope.
![](./articles_imgs/1055/slope-75.gif)
Example 3.
Given points: (3, 2), (6, - 2) Find dy and dx directly from the table or the points.
Either order gives the same slope.
![](./articles_imgs/1055/slope-76.gif)
If we are given two points (a, b) and (x, y), and the slope m and we can use the slope- form
of the equation for the line:
![](./articles_imgs/1055/slope-77.gif) |
![](./articles_imgs/1055/slope-78.gif) |
![](./articles_imgs/1055/slope-79.gif) |
![](./articles_imgs/1055/slope-80.gif) |
Multiply each side by (x - a) |
We can use this form to find the missing value
of two points (a, b), (c, d) if we know three of them.
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Example 4.
Find x if the line through (3,3) and (x,9) has the slope of 2.
![](./articles_imgs/1055/slope-81.gif)
![](./articles_imgs/1055/slope-82.gif) |
![](./articles_imgs/1055/slope-83.gif) |
Check: |
6 = 2 · (x - 3) |
Multiply each side by (x - 3) |
![](./articles_imgs/1055/slope-84.gif) |
6 = 2x - 6 |
Distributive property |
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12 = 2x |
Addition property |
![](./articles_imgs/1055/slope-85.gif) |
6 = x |
Multiplication property |
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