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Slope of a Straight Line (page 1 of 2)

Pick two x's and solve for each corresponding y: If, say, x = 3, then y = ( ²/₃ )(3) – 4 = 2 – 4 = –2. If, say, x = 9, then y = ( ²/₃ )(9) – 4 = 6 – 4 = 2. (By the way, I picked the x-values to be multiples of three because of the fraction. It's not a rule that you have to do that, but it's a helpful technique.) So the two points (3, –2) and (9, 2) are on the line y = ( ²/₃ )x – 4. To find the slope, you use the following formula:

(Why "m" for "slope", rather than, say, "s"? The official answer is: Nobody knows.)

The subscripts in the formula above just indicate that there is a first point and a second point (that is, that there are two points); it is entirely up to you which point you label as "first" and which you label as "second". The important thing is that you subtract the x's and y's in the same order. For our two points, we get the following:

Note that we could have done the points in the opposite order, and we would have come up with the exact same answer:

As you can see, the order in which you list the points really doesn't matter, as long as you subtract the x-values in the same order as you subtracted the y-values. Because of this, the slope formula can be written as it is above, or alternatively it might be written as:

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Let me emphasize: it doesn't matter which formula you use or which point is "first" and which is "second". It only matters that you subtract that x-values in the same order as you subtracted the y-values. This is true because y₁ – y₂ = –y₂ + y₁ = –(y₂ – y₁) and x₁ – x₂ = –x₂ + x₁ = –(x₂ – x₁), so doing the subtraction in the so-called "wrong" order serves only to create two "minus" signs which cancel out. The upshot: Don't worry too much about which point is the "first" point, because it really doesn't matter.

Let's find the slope of another line equation:

• Find the slope of  y = –2x + 3.  

Graphing the line, it looks like this:

I'll pick a couple values for x, and find the values for y. Picking x = –1, I get y = –2(–1) + 3
= 2 + 3 = 5. Picking x = 2, I get y = –2(2) + 3 = –4 + 3 = –1. Then the points (–1, 5) and
(2, –1) are on the line y = –2x + 3. The slope of the line is then calculated as:

Now YOU try it!

Now go back and look at those equations and their graphs. For the first one, y = ( ²/₃ )x – 4, the slope was m = ²/₃. Also, the line, as you move from left to right along the x-axis, is going up; technically, "the line is increasing". For the second line, y = –2x + 3, the slope was m = –2. Also, the line, as you move from left to right along the x-axis, is going down; technically, "the line is decreasing". This relationship is always true! For a straight line, the number on x is always the slope, and positive slopes go with increasing lines, and negative slopes go with decreasing lines. Always!

By the way, this can help you when you're doing calculations: if you calculate a slope as being negative, but you can see from the picture (graph) that the line is increasing (so the slope must be positive), you know you need to re-do your calculations. Seeing this relationship can save you points on a test, by enabling you to check your work.

Is the horizontal line going up; that is, is it an increasing line? No, so its slope won't be positive. Is the horizontal line going down; that is, is it a decreasing line? No, so its slope won't be negative. What number is neither positive nor negative? Zero! So the slope of this horizontal line is zero. Let's do the calculations: using, say, the points (–3, 4) and (5, 4), the slope is:

This relationship is true for every horizontal line: a slope of zero means the line is horizontal, and a horizontal line means you'll get a slope of zero. (By the way, all horizontals are of the form "y = a number", and "y = a number" is always a horizontal line.)

Verdict: vertical lines have NO SLOPE. That is, the concept of slope just doesn't work for vertical lines. The slope doesn't exist! Let's do the calculations. I'll use the points (4, 5) and (4, –3); the slope is:

This relationship is always true: a vertical line will have no slope, and "slope is undefined" means that the line is vertical. (By the way, all verticals are of the form "x = a number", and "x = a number" means the line is vertical. And any time your line involves an undefined slope, the line is vertical, and any time the line is vertical, you'll end up dividing by zero if you try to compute the slope.)

It is very common to confuse these two lines and their slopes, but they are very different. Just as "horizontal" is not at all the same as "vertical", so also "zero slope" is not at all the same as "no slope". "Zero" exists, so horizontal lines do indeed have a slope. But vertical lines don't have any slope; "slope" just doesn't have any meaning for vertical lines. It is very common for tests to contain questions regarding horizontals and verticals. Don't mix these up!

Now YOU try it!

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