Finding the Slope of a Line

 

Learning Objective(s)

·         Find the slope of a line from a graph.

·         Find the slope of a line given two points.

·         Find the slope of the lines x = a and y = b.

 

The idea of slope is something you encounter often in everyday life. Think about rolling a cart down a ramp or climbing a set of stairs. Both the ramp and the stairs have a slope. You can describe the slope, or steepness, of the ramp and stairs by considering horizontal and vertical movement along them. In conversation, you use words like “gradual” or “steep” to describe slope. Along a gradual slope, most of the movement is horizontal. Along a steep slope, the vertical movement is greater.

The mathematical definition of slope is very similar to our everyday one. In math, slope is used to describe the steepness and direction of lines. By just looking at the graph of a line, you can learn some things about its slope, especially relative to other lines graphed on the same coordinate plane. Consider the graphs of the three lines shown below:

 

 

First, let’s look at lines A and B. If you imagined these lines to be hills, you would say that line B is steeper than line A. Line B has a greater slope than line A.

 

Next, notice that lines A and B slant up as you move from left to right. We say these two lines have a positive slope. Line C slants down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the rise and the run. The vertical change between two points is called the rise, and the horizontal change is called the run. The slope equals the rise divided by the run: .

 

 

 

 

You can determine the slope of a line from its graph by looking at the rise and run. One characteristic of a line is that its slope is constant all the way along it. So, you can choose any 2 points along the graph of the line to figure out the slope. Let’s look at an example.

 

 

 

This line will have a slope of  no matter which two points you pick on the line. Try measuring the slope from the origin, (0, 0), to the point (6, 3). You will find that the rise = 3 and the run = 6. The slope is . It is the same!

 

Let’s look at another example.

 

 

When you look at the two lines, you can see that the blue line is steeper than the red line. It makes sense the value of the slope of the blue line, 4, is greater than the value of the slope of the red line, . The greater the slope, the steeper the line.

 

The next example shows a line with a negative slope.

 

 

 

Direction is important when it comes to determining slope. It’s important to pay attention to whether you are moving up, down, left, or right; that is, if you are moving in a positive or negative direction. If you go up to get to your second point, the rise is positive. If you go down to get to your second point, the rise is negative. If you go right to get to your second point, the run is positive. If you go left to get to your second point, the run is negative. In the example above, you could have found the slope by starting at point B, running −2, and then rising +3 to arrive at point A. The result is still a slope of .

 

 

 

 

The slope of a line can sometimes be quickly determined from its equation. Let’s consider the line whose equation is y = 5x. You can create a table of values to find 3 points on the line.

 

 

Plotting these points, create the graph of the line and determine the slope.

 

 

As you move from the point (-1, -5) to the point (2, 10), the line has a rise of 15 and a run of 3, so the slope of the line is . Notice that the number 5 also appears in the equation: y = 5x.

 

Whenever the equation of a line is written in the form y = mx + b, it is called the slope-intercept form of the equation. The m is the slope of the line. And b is the b in the point that is the y-intercept (0, b).

 

For example, for the equation y = 3x – 7, the slope is 3, and the y-intercept is (0, −7).

 

What if the equation is written as 2y = 5x + 1? Then you must rewrite the equation in the form y = mx + b. Solve for y.

 

            2y = 5x + 1

              y =  divide both sides of the equation by 2.

 

The slope is, and the y-intercept is (0, ).

 

What is the slope of the line whose equation is y = −2x + 7? A) 7 B) 2 C) −2 D)   Show/Hide Answer A) 7 Incorrect. The slope for a line written in y = mx + b is given by the coefficient of x. The correct answer is −2.   B) 2 Incorrect. The slope for a line written in y = mx + b is given by the coefficient of x. The coefficient is −2. The correct answer is −2.   C) −2 Correct. The slope for a line written in y = mx + b is given by the coefficient of x. For this line the coefficient, or m, the slope, is −2.   D) Incorrect. The slope for a line written in y = mx + b is given by the coefficient of x. The coefficient is −2. The correct answer is −2.

 

 

You’ve seen that you can find the slope of a line on a graph by measuring the rise and the run. You can also find the slope of a straight line without its graph if you know the coordinates of any two points on that line. Every point has a set of coordinates: an x-value and a y-value, written as an ordered pair (x, y). The x value tells you where a point is horizontally. The y value tells you where the point is vertically.

 

Consider two points on a line—Point 1 and Point 2. Point 1 has coordinates (x₁, y₁) and Point 2 has coordinates (x₂, y₂).

 

 

The rise is the vertical distance between the two points, which is the difference between their y-coordinates. That makes the rise y₂ − y₁. The run between these two points is the difference in the x-coordinates, or x₂ − x₁.

 

So,  or  

In the example below, you’ll see that the line has two points each indicated as an ordered pair. The point (0, 2) is indicated as Point 1, and (−2, 6) as Point 2. So you are going to move from Point 1 to Point 2. A triangle is drawn in above the line to help illustrate the rise and run.

 

 

You can see from the graph that the rise going from Point 1 to Point 2 is 4, because you are moving 4 units in a positive direction (up). The run is −2, because you are then moving in a negative direction (left) 2 units. Using the slope formula, .

 

You do not need the graph to find the slope. You can just use the coordinates, keeping careful track of which is Point 1 and which is Point 2. Let’s organize the information about the two points:

 

 

The slope,  = . The slope of the line, m, is −2.

 

It doesn’t matter which point is designated as Point 1 and which is Point 2. You could have called (−2, 6) Point 1, and (0, 2) Point 2. In that case, putting the coordinates into the slope formula produces the equation . Once again, the slope m = −2. That’s the same slope as before. The important thing is to be consistent when you subtract: you must always subtract in the same order y₂ − y₁ and x₂ − x₁.

 

 

The example below shows the solution when you reverse the order of the points, calling (5, 5) Point 1 and (4, 2) Point 2.

 

 

Notice that regardless of which ordered pair is named Point 1 and which is named Point 2, the slope is still 3.

 

 

 

What is the slope of a line that includes the points (−5, 1) and (−2, 3)   A) B) C) D)   Show/Hide Answer A) Correct.   B) Incorrect. The denominator is −2− (−5), not −2 − 5. The correct answer is .   C) Incorrect. Put the coordinates into the slope formula consistently: .  The correct answer is .   D) Incorrect. You have interchanged the rise and the run. The correct answer is .

 

 

Advanced Question What is the slope of a line that includes the points  and ?   A) B) C) D)   Show/Hide Answer A) Incorrect. It looks like you inverted the rise and the run. Use the formula  to find the slope. The correct answer is .   B) Incorrect. It looks like you subtracted either the y or x coordinates in the wrong order. Make sure you subtract , then , and then calculate the slope. The correct answer is .   C) Incorrect. It looks like you subtracted either the y or x coordinates in the wrong order. Make sure you subtract , then , and then calculate the slope. The correct answer is .   D) Correct. Using the formula for slope, , you found that .

 

So far you’ve considered lines that run “uphill” or “downhill.” Their slopes may be steep or gradual, but they are always positive or negative numbers. But there are two other kinds of lines, horizontal and vertical. What is the slope of a flat line or level ground? Of a wall or a vertical line?

 

Let’s consider a horizontal line on a graph. No matter which two points you choose on the line, they will always have the same y-coordinate. The equation for this line is y = 3. The equation can also be written as y = (0)x + 3.

 

 

Using the form y = 0x + 3, you can see that the slope is 0. You can also use the slope formula with two points on this horizontal line to calculate the slope of this horizontal line. Using (−3, 3) as Point 1 and (2, 3) as Point 2, you get:

 

 

The slope of this horizontal line is 0.

 

Let’s consider any horizontal line. No matter which two points you choose on the line, they will always have the same y-coordinate. So, when you apply the slope formula, the numerator will always be 0. Zero divided by any non-zero number is 0, so the slope of any horizontal line is always 0.

 

The equation for the horizontal line y = 3 is telling you that no matter which two points you choose on this line, the y-coordinate will always be 3.

 

How about vertical lines? In their case, no matter which two points you choose, they will always have the same x-coordinate. The equation for this line is x = 2.

 

 

There is no way that this equation can be put in the slope-point form, as the coefficient of y is 0 (x = 0y + 2).

 

So, what happens when you use the slope formula with two points on this vertical line to calculate the slope? Using (2, 1) as Point 1 and (2, 3) as Point 2, you get:

 

 

But division by zero has no meaning for the set of real numbers. Because of this fact, it is said that the slope of this vertical line is undefined. This is true for all vertical lines— they all have a slope that is undefined.

 

 

 

Advanced Question Which of the following points will lie on the line created by the points  and ?   A) B) C) D)   Show/Hide Answer A) Incorrect. Notice that both points on the line have the same x-coordinate but different y-coordinates. That makes it a vertical line, so any other points on the line will have an x-coordinate of -3.75. The correct answer is .   B) Correct.  The points  and  form a vertical line, so any other point on that line will have to have an x-coordinate of -3.75.   C) Incorrect. Try drawing a quick sketch of the points  and . They form a vertical line, so any other points on the line will have an x-coordinate of -3.75. The correct answer is .   D) Incorrect. Notice that both points on the line have the same x-coordinate but different y-coordinates. That makes it a vertical line, so any other points on the line will have an x-coordinate of -3.75. The correct answer is .

 

 

Slope describes the steepness of a line. The slope of any line remains constant along the line. The slope can also tell you information about the direction of the line on the coordinate plane. Slope can be calculated either by looking at the graph of a line or by using the coordinates of any two points on a line. There are two common formulas for slope: Slope =  and  where m = slope and  and  are two points on the line.

 

The images below summarize the slopes of different types of lines.