Slope of the line

One of the most important properties of a straight line is in how it angles away from the horizontal. This concept is reflected in something called the "slope" of the line.
Let's take a look at the straight line y = ( ²/₃ ) x – 4. Its graph looks like this:

To find the slope, we will need two points from the line.
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 merely indicate that you have a "first" point (whose coordinates are subscripted with a "1") and a "second" point (whose coordinates are subscripted with a "2"); that is, the subscripts indicate nothing more than the fact that you have two points to work with. It is entirely up to you which point you label as "first" and which you label as "second". For computing slopes with the slope formula, the important thing is that you subtract the x's and y's in the same order. For our two points, if we choose (3, –2) to be the "first" point, then we get the following:

The first y-value above, the –2, was taken from the point (3, –2) ; the second y-value, the 2, came from the point (9, 2); the x-values 3 and 9 were taken from the two points in the same order. If we had taken the coordinates from the points in the opposite order, the result would have been exactly the same value:

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 can be written as:

Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

Let me emphasize: it does not matter which of the two formulas you use or which point you pick to be "first" and which you pick to be "second". The only thing that matters is that you subtract your x-values in the same order as you had subtracted your y-values.

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Technically, the equivalence of the two slope formulas above can be proved by noting that:

y₁ – y₂ = –y₂ + y₁ = –(y₂ – y₁)
x₁ – x₂ = –x₂ + x₁ = –(x₂ – x₁)

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. (And please don't send me an e-mail claiming that the order does somehow matter, or that one of the above two formulas is somehow "wrong". If you think I'm wrong, plug pairs of points into both formulas, and try to prove me wrong! And keep on plugging until you "see" that the mathematics is in fact correct.)

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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 of values for x, and find I'll find the corresponding 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!

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Scroll back up this page and look at those equations and their graphs. For the first equation, y
= ( ²/₃ )x – 4, the slope was m = ²/₃. And the line, as you moved from left to right along the x-axis, was heading up toward the top of the drawing; technically, the line was "increasing". For the second line, y = –2x + 3, the slope was m = –2. And the line, as you moved from left to right along the x-axis, was heading down toward the bottom of the drawing; technically, the line was "decreasing". This relationship is always true: Increasing lines have positive slopes, and decreasing lines have negative slopes. Always!
This fact can help you check your calculations: if you calculate a slope as being negative, but you can see from the graph that the line is increasing (so the slope must be positive), you know you need to re-do your calculations. Being aware of this connection can save you points on a test because it will enable you to check your work before you hand it in.

Increasing lines have positive slopes; decreasing lines have negative slopes. With this in mind, consider the following horizontal line: y = 4 Its graph is shown to the right.

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 to confirm this value. Using 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 horizontal lines are of the form "y = some number", and the equation "y = some number" always graphs as a horizontal line.)

Now consider the vertical line x = 4: Is the vertical line going up on one end? Well, kind of. Is the vertical line going down on the other end? Well, kind of. Is there any number that is both positive and negative? Nope.

Verdict: vertical lines have NO SLOPE. In particular, the concept of slope simply does not 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:

(We can't divide by zero, which is of course why this slope value is "undefined".)
This relationship is always true: a vertical line will have no slope, and "the slope is undefined" means that the line is vertical. (By the way, all vertical lines are of the form "x = some number", and "x = some number" means the line is vertical. 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.)
Warning: 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". The number "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 them up!
Parallel lines and their slopes are easy. Since slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope — and lines with the same slope are parallel.

Perpendicular lines are a bit more complicated. If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will be a decreasing line). So perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down. Put this together with the sign change, and you get that the slope of the perpendicular line is the "negative reciprocal" of the slope of the original line — and two lines with slopes that are negative reciprocals of each other are perpendicular to each other. In numbers, if the one line's slope is m = ⁴/₅, then the perpendicular line's slope will be m = ^–5/₄. If the one line's slope is m = –2, then the perpendicular line's slope will be m = ¹/₂.
In your homework, you will probably be given some pairs of points, and be asked to state whether the lines through the pairs of points are "parallel, perpendicular, or neither". To answer the question, you'll have to calculate the slopes and compare them. Here's how that works:

• One line passes through the points (–1, –2) and (1, 2); another line passes through the points (–2, 0) and (0, 4). Are these lines parallel, perpendicular, or neither?

To answer this question, I'll find the slopes.

Since these two lines have identical slopes, then these lines are parallel.

• One line passes through the points (0, –4) and (–1, –7); another line passes through the points (3, 0) and (–3, 2). Are these lines parallel, perpendicular, or neither?

I'll find the values of the slopes. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

 
If I were to flip the "3" and then change its sign, I would get " ^–1/₃". In other words, these slopes are negative reciprocals, so the lines through the points are perpendicular.

• One line passes through the points (–4, 2) and (0, 3); another line passes through the points (–3, –2) and (3, 2). Are these lines parallel, perpendicular, or neither?

I'll find the slopes.

 
These slope values are not the same, so the lines are not parallel. The slope values are not negative reciprocals either, so the lines are not perpendicular. Then the answer is "neither".

Warning: When asked a question of this type ("are they parallel or perpendicular?"), do not start drawing pictures. If the lines are close to being parallel or close to being perpendicular (or if you draw the lines messily), you can very-easily get the wrong answer from your picture. Besides, they're not asking if the lines look parallel or perpendicular; they're asking if the lines actually are parallel or perpendicular. To be sure of your answer, do the algebra.