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.
Introduction
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.
Defining Slope
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.
Example  
Problem  Use the graph to find the slope of the line.
 
 rise = 2  Start from a point on the line, such as (2, 1) and move vertically until in line with another point on the line, such as (6, 3). The rise is 2 units. It is positive as you moved up. 
 run = 4  Next, move horizontally to the point (6, 3). Count the number of units. The run is 4 units. It is positive as you moved to the right. 
 Slope = _{}  Slope = _{}. 
Answer  The slope is _{}. 

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.
Example  
Problem  Use the graph to find the slope of the two lines.

 

 Notice that both of these lines have positive slopes, so you expect your answers to be positive.  

rise = 4  Blue line
Start with the blue line, going from point (2, 1) to point (1, 5). This line has a rise of 4 units up, so it is positive.  
 run = 1  Run is 1 unit to the right, so it is positive.  
 Slope = _{}  Substitute the values for the rise and run in the formula Slope = _{}.  

rise = 1  Red line
The red line, going from point (1, 2) to point (3, 1) has a rise of 1 unit.  
 run = 4  The red line has a run of 4 units.  
 Slope = _{}  Substitute the values for the rise and run into the formula Slope =_{}.  
Answer  The slope of the blue line is 4 and the slope of the red line is _{}. 
 
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.
Example  
Problem  Find the slope of the line graphed below.
 
 rise = −3  Start at Point A, (0, 4) and rise −3.This means moving 3 units in a negative direction. 
 run = 2  From there, run 2 units in a positive direction to Point B (2, 1). 
 Slope = _{}  Slope = _{}. 
Answer  The slope of the line is _{}. 
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 _{}.
Advanced Example  
Problem  Find the slope of the line graphed below.
 
rise = 4.5  Start at (3, 0.25) and rise 4.5. This means moving 4.5 units in a positive direction.  
run = 6  From there, run 6 units in a positive direction to (3, 4.25).  
_{}  _{}  
Answer  The slope of the line is 0.75.  
Looking at Equations
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.
x  y 
−1  −5 
0  0 
2  10 
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 slopeintercept form of the equation. The m is the slope of the line. And b is the b in the point that is the yintercept (0, b).
For example, for the equation y = 3x – 7, the slope is 3, and the yintercept 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 yintercept is (0, _{}).
What is the slope of the line whose equation is y = −2x + 7? A) 7 B) 2 C) −2 D) _{}

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 xvalue and a yvalue, 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 (x_{1}, y_{1}) and Point 2 has coordinates (x_{2}, y_{2}).
The rise is the vertical distance between the two points, which is the difference between their ycoordinates. That makes the rise y_{2} − y_{1}. The run between these two points is the difference in the xcoordinates, or x_{2} − x_{1}.
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:
Name  Ordered Pair  Coordinates 
Point 1  (0, 2)  x_{1} = 0 y_{1} = 2 
Point 2  (−2, 6)  x_{2} = 2 y_{2} = 6 
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 y_{2} − y_{1 }and x_{2} − x_{1}.
Example  
Problem  What is the slope of the line that contains the points (5, 5) and (4, 2)?  
 x_{1 }= 4 y_{1 }= 2  (4, 2) = Point 1, (x_{1}, y_{1}) 
 x_{2 }= 5 y_{2 }= 5  (5, 5) = Point 2, (x_{2}, y_{2}) 
 _{} _{} m = 3  Substitute the values into the slope formula and simplify. 
Answer  The slope is 3. 
The example below shows the solution when you reverse the order of the points, calling (5, 5) Point 1 and (4, 2) Point 2.
Example  
Problem  What is the slope of the line that contains the points (5, 5) and (4, 2)?  
 x_{1 }= 5 y_{1 }= 5  (5, 5) = Point 1, (x_{1}, y_{1}) 
 x_{2 }= 4 y_{2 }= 2  (4, 2) = Point 2, (x_{2}, y_{2}) 
 _{} _{} m = 3  Substitute the values into the slope formula and simplify. 
Answer  The slope is 3. 
Notice that regardless of which ordered pair is named Point 1 and which is named Point 2, the slope is still 3.
Advanced Example  
Problem  What is the slope of the line that contains the points (3,6.25) and (1,8.5)?  
_{}  (3,6.25) = Point 1, _{}  
_{}  (1,8.5) = Point 2, _{}  
_{} 
Substitute the values into the slope formula and simplify.  
Answer  The slope is 3.6875. 

What is the slope of a line that includes the points (−5, 1) and (−2, 3)
A) _{} B) _{} C) _{} D) _{}

Advanced Question What is the slope of a line that includes the points _{} and _{}?
A) _{} B) _{} C) _{} D) _{}

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 ycoordinate. 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 ycoordinate. So, when you apply the slope formula, the numerator will always be 0. Zero divided by any nonzero 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 ycoordinate will always be 3.
How about vertical lines? In their case, no matter which two points you choose, they will always have the same xcoordinate. The equation for this line is x = 2.
There is no way that this equation can be put in the slopepoint 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.
Example  
Problem  What is the slope of the line that contains the points (3, 2) and (−8, 2)?  
 _{} _{}  (3, 2) = Point 1, _{} 
 _{} _{}  (−8, 2) = Point 2, _{} 
 _{}
_{} m = 0  Substitute the values into the slope formula and simplify. 
Answer  The slope is 0, so the line is horizontal. 
Advanced Question Which of the following points will lie on the line created by the points _{} and _{}?
A) _{} B) _{} C) _{} D) _{}

Summary
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.