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Line Slope Value
> ... Math > Slopes, Intercepts, Equation of a Straight Line > Line Slope Value

Calculating the value of the slope is important in defining a function. This lesson will teach you how to find the value of the slope. Slope is usually represented by the variable m.

The Value of the Slope of a Line

Video Source (08:42 mins) | Transcript

Zero and Undefined Slope

Video Source (02:24 mins) | Transcript

\( Slope = m = {\frac {rise}{run}} \)

  • m = the variable we use for slope
  • Rise = how far up your step goes
  • Run = how far over your step goes

Remember to check to see if your slope should be positive or negative.

Additional Resources

Practice Problems

Use the following graph to answer questions 1 through 7.
This is an image of a graph with five different lines. Line A starts in the top left and ends in the bottom right. Line B starts in the bottom left and ends in the top right. Line C starts in the top left and ends in the bottom right. Line D starts around the lower middle area of the left side and ends in the upper middle area of the right side. Line E starts around the x axis on the left and ends 4 units down from the x axis on the right.
  1. What is the slope of line A? (
    Solution
    x
    Solution: \(\text{m} = -\dfrac{7}{2}\)
    Details:

    Step 1: Locate two points on the graph that will be used to find the slope. It is best to choose points with x and y values that are integers, if possible. We will choose \((−2,3)\) and \((0,−4)\).

    Step 2: Draw a vertical line and a horizontal line to represent the ‘step’ between the two points:
    A coordinate plane with several lines graphed. The line marked A passes through the point (negative 2,3) and (0, negative 4). There is a dashed vertical line from the point (negative 2, 3) to the point (negative 2, negative 4) and a dashed horizontal line from the point (negative 2, negative 4) to the point (0, negative 4).

    Step 3: Find the rise (or change in y value) and the run (or change in x value).
    This is the same as the previous image except there are markings indicating that the vertical line from point (negative 2, 3) to (negative 2, negative 4) represents the rise of the line which is negative 7 because the line drops down 7 units. There are also markings to show that the horizontal line from the point (negative 2, negative 4) to the point (0, negative 4) represents the run of the line which is 2 because the line goes to the right 2 units.

    Step 4: Find the slope. The slope is the change in y value (or how much the line went up or down between the two points) divided by the change in x value (or how much the step moves to the right). The slope is negative since it goes down when we follow the line from left to right. The slope is:

    \({\text{m=}}\dfrac{\text{change in y}}{\text{change in x}}=-\dfrac{7}{2}\)

    \({\text{m}}=-\dfrac{7}{2}\)
    )
  2. What is the slope of line B? (
    Solution
    x
    Solution:
    \(\text{m}=1\)
    )
  3. What is the slope of line C? (
    Solution
    x
    Solution:
    \(\text{m}=-2\)
    )
  4. What is the slope of line D? (
    Solution
    x
    Solution:
    \(\text{m} = \dfrac{1}{3}\)
    )
  5. What is the slope of line E? (
    Solution
    x
    Solution:
    \(\text{m} = -\dfrac{2}{5}\)
    )
  6. What is the slope of a horizontal line? (
    Solution
    x
    Solution:
    \(\text{m} = 0\)
    )
  7. What is the slope of a vertical line? (
    Solution
    x
    Solution: The slope of a vertical line is undefined.
    Details:

    Step 1: Locate two points on the graph that will be used to find the slope. Since we don’t have a graph of this line we’ll draw a sample line and include two points so that we can find the slope:
    This is a picture of a coordinate plane with a vertical line graphed through the point (1,0) and points marked on the line at (1,0) and (1, 2).

    Step 2: Draw a vertical line and a horizontal line to represent the ‘step’ between the two points. In this case, we can only draw a vertical line because there is no change in the x values:
    This is a picture of a coordinate plane with a vertical line graphed through the point (1,0) and points marked on the line at (1,0) and (1, 2). There is a dashed line from the point (1,0) and (1,2) showing the rise of the line. Rise = 2 and Run = 0 is written on the graph.

    Step 3: Find the rise (or change in y-value) and the run (or change in x-value). The rise, or change in y-value, is 2. There is no change in the x-value so the run is 0.

    Step 4: Find the slope. The slope is the change in y-value (or how much the line went up or down between the two points) divided by the change in x-value (or how much the step moved to the right).

    \({\text{m}}=\dfrac{\text{change in y}}{\text{change in x}}=\dfrac{2}{0}\)

    So m = undefined because we can’t divide by zero.
    )

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    Introduction to Slopes
    X and Y Intercepts of a Line