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Slope

When you connect a line between two points it seems to have a slope, or a certain amount of rise for each unit that it runs horizontally.  Just like the slope of a hill, the slope of a line in mathematics is defined the way you think it should be.  Here is the formal definition:

Definition:  Let \(\left( {{x_1},{y_1}} \right)\) and \(\left( {{x_2},{y_2}} \right)\) be any two points in the plane.  Then the slope \(m\) between the two points is

\(m = \Large \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\)

One way to think of slope is as “rise” over “run,” but if you just memorize the definition, you will do well throughout your mathematical career.  Now let’s see if we can find the slope between two points.

Example:  Find the slope of the line through the points \(A\left( {10,9} \right)\) and \(B\left( {18, - 1} \right)\).

Solution:  We use the definition of slope, with \({x_1} = 10\), \({x_2} = 18\), \({y_1} = 9\) and \({y_2} =  - 1\):

\(\begin{gathered}
  m = \large \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}  \\
   = \large \frac{{ - 1 - 9}}{{18 - 10}}  \\
   = \large \frac{{ - 10}}{8}  \\
   =  - \large \frac{5}{4}  \\
\end{gathered} \)

Visually, this understanding of slope makes sense, especially the sense of “rise” over “run”, as the graph below will show.  Notice that it takes two occurrences of the slope \( - \Large \frac{5}{4}\) to get from point \(A\) to point \(B\), but only one occurrence of the slope \( - \Large \frac{{10}}{8}\), which is an equivalent fraction.

Linear-Equations-and-Inequalities-Slope-1

Let’s finish off with one more example:

Example:  Find the slope of the line between the points \(\left( { - 15,18} \right)\) and \(\left( {0,5} \right)\).

Solution:  Again we use the definition of slope, with \({x_1} =  - 15\), \({x_2} = 0\), \({y_1} = 18\), and \({y_2} = 5\). We have

\(\begin{gathered}
  m = \large \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}  \\
   = \large \frac{{5 - 18}}{{0 - \left( { - 15} \right)}}  \\
   = \large \frac{{ - 13}}{{15}}  \\
\end{gathered} \)

 

Below you can download some free math worksheets and practice.


Downloads:
2235 x

Find the slope of each line.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Linear-Equations-and-Inequalities-Slope-easy

Watch below how to solve this example:

 

Downloads:
1650 x

Find the slope of the line through each pair of points.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Linear-Equations-and-Inequalities-Slope-medium

Watch below how to solve this example:

 

Downloads:
1857 x

Find the slope of each line.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Linear-Equations-and-Inequalities-Slope-hard

Watch below how to solve this example:

 
 
 

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