Slope Formula and Point-Slope Formula


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Posted by dolphin@eskimo.com (64.24.35.67) on August 14, 2002 at 10:28:28:

In Reply to: HELP me help my mom with algebra posted by on August 13, 2002 at 17:25:04:


I'm going to cut & paste a previous response on the same subject. See if you can follow it; then try working through your own problem.

If you're still stuck, then let us know.

~ Mark

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A linear relationship between x and y means that the graph is a straight line. In order to write the equation of a straight line, you need to know the line's slope. (The slope is a number that gives some indication of how steep the line is.)

Two points are sufficient to define a line. If you're given the coordinates of any two points on the line, then you can calculate the slope by using a formula (given below). If you're only given the coordinates of one point, then you need to be given the slope also because there are an infinite number of potential lines that pass through that one point.

Given the coordinates of two points: (x1,y1) and (x2,y2)

slope = (y1 - y2) / (x1 - x2)

So, the slope is the difference of the y-coordinates divided by the difference of the x-coordinates.

Here are some examples (the letter "m" is usually chosen to represent the slope):

Given two points: (2,7) and (3,4)

m = (7 - 4) / (2 - 3)

m = 3 / -1

m = -3

So, the slope of the line connecting the points (2,7) and (3,4) is -3. It doesn't matter which point you choose to be (x1,y1). It will work either way.

Given two points: (1,8) and (-3,5)

m = (5 - 8) / (-3 - 1)

m = -3 / -4

m = 3/4

Now that you know how to calculate a slope, you are ready to write the equation. We will use something called the "point-slope" formula. It is called the point-slope formula because you need to know the coordinates of one point and the slope to use it.

Given the coordinates of one point (x1,y1), and knowing the slope m, use the following formula to write the equation:

y - y1 = m (x - x1)


Let's write the equation of the line passing through the point (5,9) with slope 4. So, x1 = 5, y1 = 9, and m = 4.

y - 9 = 4 (x - 5)

y - 9 = 4x - 20

y = 4x - 11


Let's write the equation of the line passing through the points (2,7) and (3,4). We already calculated the slope of this line in my first example; m = -3. You can pick either point to be (x1,y1). I pick (2,7).

y - 7 = -3 (x - 2)

y - 7 = -3x + 6

y = -3x + 13


Let's look at one last example.

Write the equation of the line passing through the points (9,-17) and (-4,4).

Calculate the slope:

m = (-17 - 4) / (9 + 4)

m = -21 / 13

Use the point-slope formula:

y - 4 = (-21/13)(x + 4)

y - 4 = (-21/13)x - 84/13

y = (-21/13)x - 32/13



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