Slope of a line

Slope of a Straight Line

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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!
Slope: Parallel and Perpendicular Lines

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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.

x- and y-Intercepts

x- and y-Intercepts

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The graphical concept of x- and y-intercepts is pretty simple. The x-intercepts are where the graph crosses the x-axis, and the y-intercepts are where the graph crosses the y-axis. The problems start when we try to deal with intercepts algebraically.
To clarify the algebraic part, think again about the axes. When you were first introduced to the Cartesian plane, you were shown the regular number line from elementary school (the x-axis), and then shown how you could draw a perpendicular number line (the y-axis) through the zero point on the first number line. Take a closer look, and you'll see that the y-axis is also the line "x = 0". In the same way, the x-axis is also the line "y = 0".
Then, algebraically,

• an x-intercept is a point on the graph where y is zero, and
• a y-intercept is a point on the graph where x is zero.

More specifically,

• an x-intercept is a point in the equation where the y-value is zero, and
• a y-intercept is a point in the equation where the x-value is zero.
     

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• Find the x- and y-intercepts of 25x² + 4y² = 9

Using the definitions of the intercepts, I will proceed as follows:
x-intercept(s):

y = 0 for the x-intercept(s), so:

25x² + 4y² = 9
25x² + 4(0)² = 9
25x² + 0 = 9
x² =  ⁹/₂₅
x = ± ( ³/₅ )
Then the x-intercepts are the points ( ³/₅, 0) and ( ^–3/₅, 0)
y-intercept(s):   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

x = 0 for the y-intercept(s), so:

25x² + 4y² = 9
25(0)² + 4y² = 9
0 + 4y² = 9
y² =  ⁹/₄
y = ± ( ³/₂ )
Then the y-intercepts are the points (0, ³/₂ ) and (0, ^–3/₂ )

Just remember: Whichever intercept you're looking for, the other variable gets set to zero.

Inequalities

Solving linear inequalities is very similar to solving linear equations, except for one small but important detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative. The easiest way to show this is with some examples:

1)	Graphically, the solution is:	The only difference between the linear equation "x + 3 = 2" and this linear inequality is that I have a "less than" sign, instead of an "equals" sign. The solution method is exactly the same: subtract 3 from either side. Note that the solution to a "less than, but not equal to" inequality is graphed with a parentheses (or else an open dot) at the endpoint, indicating that the endpoint is not included within the solution.
2)	Graphically, the solution is:	The only difference between the linear equation "2 – x = 0" and this linear inequality is the "greater than" sign in place of an "equals" sign. Note that "x" in the solution does not "have" to be on the left. However, it is often easier to picture what the solution means with the variable on the left. Don't be afraid to rearrange things to suit your taste.
3)	Graphically, the solution is:	The only difference between the linear equation "4x + 6 = 3x – 5" and this inequality is the "less than or equal to" sign in place of a plain "equals" sign. The solution method is exactly the same. Note that the solution to a "less than or equal to" inequality is graphed with a square bracket (or else a closed dot) at the endpoint, indicating that the endpoint is included within the solution.
4)	Graphically, the solution is:	The solution method here is to divide both sides by a positive two.     Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved
5)	Graphically, the solution is:	This is the special case noted above.  When I divided by the negative two, I had to flip the inequality sign.

The rule for example 5 above often seems unreasonable to students the first time they see it. But think about inequalities with numbers in there, instead of variables. You know that the number four is larger than the number two: 4 > 2. Multiplying through this inequality by –1, we get –4 < –2, which the number line shows is true:

If we hadn't flipped the inequality, we would have ended up with "–4 > –2", which clearly isn't true.
The previous inequalities are called "linear" inequalities because we are dealing with linear expressions like "x – 2" ("x > 2" is just "x – 2 > 0", before you finished solving it). When we have an inequality with "x²" as the highest-degree term, it is called a "quadratic inequality". The method of solution is more complicated.

• Solve x² – 3x + 2 > 0

First, I have to find the x-intercepts of the associated quadratic, because the intercepts are where y = x² – 3x + 2  is equal to zero. Graphically, an inequality like this is asking me to find where the graph is above or below the x-axis. It is simplest to find where it actually crosses the x-axis, so I'll start there.
Factoring, I get x² – 3x + 2 = (x – 2)
(x – 1) = 0, so x = 1 or x = 2. Then the graph crosses the x-axis at 1 and 2, and the number line is divided into the intervals (negative infinity, 1), (1, 2), and (2, positive infinity). Between the x-intercepts, the graph is either above the axis (and thus positive, or greater than zero), or else below the axis (and thus negative, or less than zero).
There are two different algebraic ways of checking for this positivity or negativity on the intervals. I'll show both.
1) Test-point method. The intervals between the x-intercepts are (negative infinity, 1), (1, 2), and (2, positive infinity). I will pick a point (any point) inside each interval. I will calculate the value of y at that point. Whatever the sign on that value is, that is the sign for that entire interval.
For (negative infinity, 1), let's say I choose x = 0; then y = 0 – 0 + 2 = 2, which is positive. This says that y is positive on the whole interval of (negative infinity, 1), and this interval is thus part of the solution (since I'm looking for a "greater than zero" solution).
For the interval (1, 2), I'll pick, say, x = 1.5; then y = (1.5)² – 3(1.5) + 2 = 2.25 – 4.5 + 2 = 4.25 – 4.5 = –0.25, which is negative. Then y is negative on this entire interval, and this interval is then not part of the solution.
For the interval (2, positive infinity), I'll pick, say, x = 3; then y = (3)² – 3(3) + 2 = 9 – 9 + 2 = 2, which is positive, and this interval is then part of the solution. Then the complete solution for the inequality is x < 1 and x > 2. This solution is stated variously as:


	inequality notation
	interval, or set, notation
	number line with parentheses (brackets are used for closed intervals)
	number line with open dots (closed dots are used for closed intervals)

The particular solution format you use will depend on your text, your teacher, and your taste. Each format is equally valid.   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved
2) Factor method. Factoring, I get y = x² – 3x + 2 = (x – 2)(x – 1). Now I will consider each of these factors separately.
The factor x – 1 is positive for x > 1; similarly, x – 2 is positive for x > 2. Thinking back to when I first learned about negative numbers, I know that (plus)×(plus) = (plus), (minus)×(minus) = (plus), and (minus)×(plus) = (minus). So, to compute the sign on y = x² – 3x + 2, I only really need to know the signs on the factors. Then I can apply what I know about multiplying negatives.


First, I set up a grid, showing the factors and the number line.
Now I mark the intervals where each factor is positive.
Where the factors aren't positive, they must be negative.
Now I multiply up the columns, to compute the sign of y on each interval.

Then the solution of x² – 3x + 2 > 0 are the two intervals with the "plus" signs:

(negative infinity, 1) and (2, positive infinity).

• Solve –2x² + 5x + 12 < 0.

First I find the zeroes, which are the endpoints of the intervals: y = –2x² + 5x + 12 =
(–2x – 3)(x – 4) = 0 for x = –³/₂ and x = 4. So the endpoints of the intervals will be at –³/₂ and 4. The intervals are between the endpoints, so the intervals are (negative infinity, ^–3/₂], [^–3/₂, 4], and [4, positive infinity). (Note that I use brackets for the endpoints in "or equal to" inequalities, instead of parentheses, because the endpoints will be included in the final solution.)
To find the intervals where y is negative by the Test-Point Method, I just pick a point in each interval. I can use points such as x = –2, x = 0, and x = 5.
To find the intervals where y is negative by the Factor Method, I just solve each factor: –2x – 3 is positive for –2x – 3 > 0, –3 > 2x, –3/2 > x, or x < –³/₂; and x – 4 is positive for x – 4 > 0,
x > 4. Then I fill out the grid:

    

Then the solution to this inequality is all x's in
(negative infinity, ^–3/₂ ] and [4, positive infinity).

Solving Multi-Step Equations

Solving Multi-Step Equations

In multi-step equations, you will need to make use of the techniques used in solving one-step and two-step equations. You may want to review those topics before beginning the examples in this lesson. Just as with solving one-step or two-step or any equation, one goal in solving an equation is to have only variables on one side of the equal sign and numbers on the other side of the equal sign. The other goal is to have the number in front of the variable equal to one. Keep in mind that the variable does not always have to be x. These equations can make use of any letter as a variable. The strategy for getting the variable by itself with a coefficient of 1 involves using opposite operations. For example, to move something that is added to the other side of the equation, you should subtract. The most important thing to remember in solving a linear equation is that whatever you do to one side of the equation, you MUST do to the other side. So if you subtract a number from one side, you MUST subtract the same value from the other side. You will see how this works in the examples. Solve This problem is typical of what you can expect to see in a multi-step equation. This equation has a variable on both sides of the equal sign. We must first put the variables on the same side. Let’s move the 2x from the right side to the left side by subtracting 2x from both sides. Now that we have isolated the variable on the left side of the equation, we can go about solving the new equation using techniques of solving one and two step equations. Solve We must first make sure we have the variable only on one side. It does not matter which side we choose. Some people prefer to always move the variable to the left side while others try to make sure they don’t have a negative coefficient. There is no “always correct” method. The most important thing is to know how to get the variable only on one side. Solve Before we begin to use any of our equation solving skills, we must first simplify the equation by using the distributive property and eliminating the parentheses. Now we simplify by combining like terms. At this point, we can use our equation solving skills to find the solution to the equation.

Solving one step equations

Solving One-Step Equations

A one-step equation is as straightforward as it sounds. You will only need to perform one step in order to solve the equation. One goal in solving an equation is to have only variables on one side of the equal sign and numbers on the other side of the equal sign. The other goal is to have the number in front of the variable equal to one. The strategy for getting the variable by itself with a coefficient of 1 involves using opposite operations. For example, to move something that is added to the other side of the equation, you should subtract. The most important thing to remember in solving a linear equation is that whatever you do to one side of the equation, you MUST do to the other side. So if you subtract a number from one side, you MUST subtract the same value from the other side. You will see how this works in the examples. We’ll begin solving equations with those that only require one-step. Once you work through the examples, you may want to link to other lessons that require more steps. Let's Practice: Solve Remember the goal is to have the variable by itself on one side of the equation. In this problem, that means moving the 5 to the other side of the equation. Since the 5 is added to the variable, we move it to the other side of the equation by subtracting 5. However, if we subtract 5 from the left side of the equation, we MUST also subtract 5 from the right side. Solve It does not matter that the variable in this equation is on the right side of the equation. The position of the variable is not an issue. Remember that the goal is to have the variable on one side by itself. It does not matter which side. To get the variable by itself, we need to add 3 to both sides. Solve The variable in this equation is already on one side of the equation by itself. There is no need to add or subtract anything to both sides. However, the number in front of the variable is not 1. The -3 that is in front of the variable indicates multiplication of -3 by x. The opposite operation of multiplication is division. So we will divide both sides by -3. You should take note of the different ways of writing the answer. In the example, we divided by -3, yet wrote the answer with the negative in front of the entire fraction, not just the 3. Each of the following fractions all mean the same thing. Solve The variable in this equation is already on one side by itself, but it is divided by 3. To get rid of the 3 that is attached to the variable by division, we will perform the opposite operation which is multiplication. Notice that our variable can be any letter. It does not always have to be x. Solve Once again, the variable is on one side by itself, but is multiplied by a -3 and divided by 5. Let’s take care of each operation separately and see what happens. First we’ll get rid of the 5 by multiplying both sides by 5. Then we’ll get rid of the -3 by dividing both sides by -3. Rather than perform two separate steps of multiplying by 5 and then dividing by -3, it is possible to combine those operations into one step. In other words, we can multiply both sides by . The value is called the reciprocal of . The reciprocal of a number has the same sign, but the numerator and denominator are reversed. So what was on bottom, is now on top. And what was on top, is now on bottom. If we re-work Example 5 by using the reciprocal, you can see that it will save a step in the solution process.