6.4 Slope of a Line

Slope of a Line

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When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?

In mathematics, the ‘tilt’ of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.

In this section, we will explore the concept of slope.

Use Geoboards to Model Slope

A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.

A rubber band stretches between two pegs, from peg (1, 2) up to peg (4, 4). A dashed gray run leg goes 3 units right from (1, 2) to (4, 2), and a dashed red rise leg goes 2 units up from (4, 2) to (4, 4): slope = rise over run = 2/3. Adjustable parameters: Peg 1 across (x1) = 1, Peg 1 up (y1) = 2, Peg 2 across (x2) = 4, Peg 2 up (y2) = 4. Drag either peg to a new spot. The dashed legs show the run (gray) and the rise (red); slope = rise over run. Viewing window: x from -5.94 to 10.94, y from -2.72 to 7.72. — A rubber band stretched between two pegs models a line. It starts at the book's example — from peg (1, 2) up to peg (4, 4): a run of 3 and a rise of 2, so the slope is m = rise/run = 2/3. Drag either peg to model a different line; the dashed gray leg is the run and the dashed red leg is the rise.

We’ll start by stretching a rubber band between two pegs as shown in Figure 6.11.

A 5-by-5 geoboard shows a rubber band stretched between the peg in column 1, row 4 and the peg in column 4, row 2. The resulting line has a slope of 2/3. The ends of the band are draggable, and a readout displays the current slope. Viewing window: x from -6.44 to 10.44, y from -3.22 to 7.22. — The rubber band stretches from the peg at column 1, row 4 to the peg at column 4, row 2. This creates a right triangle with a vertical rise of 2 units and a horizontal run of 3 units, giving a slope of m = 2/3. Drag the endpoints to different pegs to see how the rise and run change the steepness of the line.

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line. — **Figure 6.11**

Doesn’t it look like a line?

Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, as shown in Figure 6.12

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle. The 1, 2 peg is the vertex of the 90 degree angle, while the line between the 1, 4 and 4, 2 pegs forms the hypotenuse of the triangle. — **Figure 6.12**

We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.

To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in Figure 6.13.

In this illustration, there are two perpendicular lines with arrows. The first line extends straight upward and is labeled “rise”. The second arrow extends straight rightward and is labeled “run”. — **Figure 6.13**

If our geoboard and rubber band look just like the one shown in Figure 6.14, the rise is 2. The rubber band goes up 2 units. (Each space is one unit.)

What is the run?

The rubber band goes across 3 units. The run is 3 (see Figure 6.14).

Interactive geoboard with 5 rows and 5 columns of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line with slope 3/4. Both ends of the band can be dragged to other pegs; a readout shows the current slope. Viewing window: x from -6.44 to 10.44, y from -3.22 to 7.22. — The book's rubber band stretches from the peg in column 1, row 5 to the peg in column 5, row 2 — a rise of 3 over a run of 4, so the slope is 3/4. Drag either end of the band to new pegs and watch the rise, the run, and the slope readout change together.

The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter $m$ .

What is the slope of the line on the geoboard in Figure 6.14?

A 5-by-5 geoboard shows a rubber band stretched between the peg in column 1, row 4 and the peg in column 3, row 3. The resulting line has a slope of 1/2. The endpoints are draggable, and a readout displays the current slope. Viewing window: x from -6.44 to 10.44, y from -3.22 to 7.22. — The rubber band stretches from (1, 4) along a run of 2 and a rise of 1 to (3, 3), creating a slope of m = 1/2. Drag the endpoints to change the rise and run while keeping the slope constant at 1/2. Watch how the dashed legs of the right triangle update to show the current vertical and horizontal changes.

$\begin{matrix} m & = & \frac{rise}{run} \\ m & = & \frac{2}{3} \end{matrix}$

The line has slope $\frac{2}{3}$ . This means that the line rises 2 units for every 3 units of run.

When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. If the rise goes up it is positive and if it goes down it is negative. The run will go from left to right and be positive.

Interactive geoboard with 5 rows and 5 columns of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line with slope -1/3. Both ends of the band can be dragged to other pegs; a readout shows the current slope. Viewing window: x from -6.44 to 10.44, y from -3.22 to 7.22. — The book's rubber band stretches from the peg in column 1, row 3 to the peg in column 4, row 4 — a rise of -1 over a run of 3, so the slope is -1/3. Drag either end of the band to new pegs and watch the rise, the run, and the slope readout change together.

Example 1

What is the slope of the line on the geoboard shown?

Show solution

Use the definition of slope: $m = \frac{rise}{run} .$

Start at the left peg and count the spaces up and to the right to reach the second peg.

**Table 6.9**
The rise is 3.	$m = \frac{3}{run}$
The run is 4.	$m = \frac{3}{4}$
	The slope is $\frac{3}{4}$ .

This means that the line rises 3 units for every 4 units of run.

Example 2

What is the slope of the line on the geoboard shown?

Show solution

Use the definition of slope: $m = \frac{rise}{run} .$

Start at the left peg and count the units down and to the right to reach the second peg.

**Table 6.10**
The rise is −1.	$= \frac{−1}{run}$
The run is 3.	$\begin{matrix} m = \frac{−1}{3} \\ m = - \frac{1}{3} \end{matrix}$
	The slope is $- \frac{1}{3}$ .

This means that the line drops 1 unit for every 3 units of run.

Notice that in Example 1 the slope is positive and in Example 2 the slope is negative. Do you notice any difference in the two lines shown in Figure 6.15(a) and Figure 6.15(b)?

The figure shows two grids of evenly spaced pegs, one labeled (a) and one labeled (b). There are 5 columns and 5 rows of pegs in each grid. In the (a) grid, a rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line. Below this grid is the slope of a line defined as m equals 3 fourths. In the (b) grid, a rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line. Below this grid is the slope of a line defined as m equals negative 1 third. — **Figure 6.15**

We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line in Figure 6.15(a) is going up; it has positive slope. The line in Figure 6.15(b) is going down; it has negative slope.

A 5-by-5 geoboard shows a rubber band stretched from the peg in column 1, row 2 to the peg in column 5, row 3. The resulting line has a slope of -1/4. The endpoints are draggable, and a readout displays the current slope. Viewing window: x from -6.44 to 10.44, y from -3.22 to 7.22. — The geoboard rubber band stretches from (1, 2) along a run of 4 and a rise of -1 to the peg at (5, 3). This creates a slope of m = rise/run = -1/4. Drag the endpoints to see how changing the vertical drop affects the steepness of the downhill line.

Example 3

Use a geoboard to model a line with slope $\frac{1}{2}$ .

Show solution

To model a line on a geoboard, we need the rise and the run.

**Table 6.11**
Use the slope formula.	$m = \frac{rise}{run}$
Replace $m$ with $\frac{1}{2}$ .	$\frac{1}{2} = \frac{rise}{run}$

So, the rise is 1 and the run is 2.

Start at a peg in the lower left of the geoboard.

Stretch the rubber band up 1 unit, and then right 2 units.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is $\frac{1}{2}$ .

Example 4

Use a geoboard to model a line with slope $\frac{−1}{4} .$

Show solution

**Table 6.12**
Use the slope formula.	$m = \frac{rise}{run}$
Replace $m$ with $\frac{−1}{4}$ .	$\frac{−1}{4} = \frac{rise}{run}$

So, the rise is $−1$ and the run is 4.

Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down.

We stretch the rubber band down 1 unit, then go to the right 4 units, as shown.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is $\frac{−1}{4}$ .

Use $m = \frac{rise}{run}$ to Find the Slope of a Line from its Graph

Now, we’ll look at some graphs on the $x y$ -coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.

To find the slope, we must count out the rise and the run. But where do we start?

We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.

Example 6

Find the slope of the line shown.

Show solution

Locate two points on the graph whose coordinates are integers.	$open parenthesis 0, 5 close parenthesis$ and $open parenthesis 3, 3 close parenthesis$
Which point is on the left?	$open parenthesis 0, 5 close parenthesis$
Starting at $open parenthesis 0, 5 close parenthesis$ , sketch a right triangle to $open parenthesis 3, 3 close parenthesis$ .
Count the rise—it is negative.	The rise is $−2$ .
Count the run.	The run is 3.
Use the slope formula.	$m = \frac{rise}{run}$
Substitute the values of the rise and run.	$m = \frac{−2}{3}$
Simplify.	$m = - \frac{2}{3}$
	The slope of the line is $- \frac{2}{3}$ .

So $y$ decreases by 2 units as $x$ increases by 3 units.

What if we used the points $open parenthesis −3, 7 close parenthesis$ and $open parenthesis 6, 1 close parenthesis$ to find the slope of the line?

The rise would be $−6$ and the run would be 9. Then $m = \frac{−6}{9}$ , and that simplifies to $m = - \frac{2}{3}$ . Remember, it does not matter which points you use—the slope of the line is always the same.

In the last two examples, the lines had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points to find the slope. In the next example, the y-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier.

Example 7

Find the slope of the line shown.

Show solution

Locate two points on the graph whose coordinates are integers.	$open parenthesis 2, 3 close parenthesis$ and $open parenthesis 7, 6 close parenthesis$
Which point is on the left?	$open parenthesis 2, 3 close parenthesis$
Starting at $open parenthesis 2, 3 close parenthesis$ , sketch a right triangle to $open parenthesis 7, 6 close parenthesis$ .
Count the rise.	The rise is 3.
Count the run.	The run is 5.
Use the slope formula.	$m = \frac{rise}{run}$
Substitute the values of the rise and run.	$m = \frac{3}{5}$
	The slope of the line is $\frac{3}{5}$ .

This means that $y$ increases 3 units as $x$ increases 5 units.

When we used geoboards to introduce the concept of slope, we said that we would always start with the point on the left and count the rise and the run to get to the point on the right. That way the run was always positive and the rise determined whether the slope was positive or negative.

What would happen if we started with the point on the right?

Let’s use the points $open parenthesis 2, 3 close parenthesis$ and $open parenthesis 7, 6 close parenthesis$ again, but now we’ll start at $open parenthesis 7, 6 close parenthesis$ .

**Table 6.13**
Count the rise.	The rise is $−3$ .
Count the run. It goes from right to left, so it is negative.	The run is $−5$ .
Use the slope formula.	$m = \frac{rise}{run}$
Substitute the values of the rise and run.	$m = \frac{−3}{−5}$
	The slope of the line is $\frac{−3}{−5}$ .

It does not matter where you start—the slope of the line is always the same.

Use the Slope Formula to find the Slope of a Line Between Two Points

Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.

We have seen that an ordered pair $(x, y)$ gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol $(x, y)$ be used to represent two different points? Mathematicians use subscripts to distinguish the points.

$open parenthesis x sub 1, y sub 1 close parenthesis, read ‘ x sub 1, y sub 1’; open parenthesis x sub 2, y sub 2 close parenthesis, read ‘ x sub 2, y sub 2’$

The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?

We will use $open parenthesis x sub 1, y sub 1 close parenthesis$ to identify the first point and $open parenthesis x sub 2, y sub 2 close parenthesis$ to identify the second point.

If we had more than two points, we could use $open parenthesis x sub 3, y sub 3 close parenthesis$ , $open parenthesis x sub 4, y sub 4 close parenthesis$ , and so on.

Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points $open parenthesis 2, 3 close parenthesis$ and $open parenthesis 7, 6 close parenthesis$ .

A line passes through the marked points (2, 3) and (7, 6), labeled (x1, y1) and (x2, y2). The right triangle has corner (2, 6). Its vertical leg from (2, 3) to (2, 6) is labeled y2 - y1, 6 - 3, and 3; its horizontal leg from (2, 6) to (7, 6) is labeled x2 - x1, 7 - 2, and 5. Labeled ticks run from 0 to 8 on both axes.

Since we have two points, we will use subscript notation, $open parenthesis 2, x sub 1, 3 y sub 1 close parenthesis$ $open parenthesis 7, 6 x sub 2, y sub 2 close parenthesis$ .

On the graph, we counted the rise of 3 and the run of 5.

Notice that the rise of 3 can be found by subtracting the y-coordinates 6 and 3.

$3 = 6 - 3$

And the run of 5 can be found by subtracting the x-coordinates 7 and 2.

$5 = 7 - 2$

We know $m = \frac{rise}{run}$ . So $m = \frac{3}{5}$ .

We rewrite the rise and run by putting in the coordinates $m = \frac{6 - 3}{7 - 2}$ .

But 6 is $y sub 2$ , the y-coordinate of the second point and 3 is $y sub 1$ , the y-coordinate of the first point.

So we can rewrite the slope using subscript notation. $m = \frac{y_{2} - y_{1}}{7 - 2}$

Also, 7 is $x sub 2$ , the x-coordinate of the second point and 2 is $x sub 1$ , the x-coordinate of the first point.

So, again, we rewrite the slope using subscript notation. $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

We’ve shown that $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ is really another version of $m = \frac{rise}{run}$ . We can use this formula to find the slope of a line when we have two points on the line.

Example 8

Use the slope formula to find the slope of the line between the points $open parenthesis 1, 2 close parenthesis$ and $open parenthesis 4, 5 close parenthesis$ .

Show solution

**Table 6.14**
We'll call $open parenthesis 1, 2 close parenthesis$ point #1 and $open parenthesis 4, 5 close parenthesis$ point #2.	$open parenthesis 1, 2 x sub 1, y sub 1 close parenthesis open parenthesis 4, 5 x sub 2, y sub 2 close parenthesis$
Use the slope formula.	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .
Substitute the values.
$y$ of the second point minus $y$ of the first point	$m = \frac{5 - 2}{x_{2} - x_{1}}$ .
$x$ of the second point minus $x$ of the first point	$m = \frac{5 - 2}{4 - 1}$ .
Simplify the numerator and the denominator.	$m = \frac{3}{3}$ .
Simplify.	$m equals 1$ .

Let’s confirm this by counting out the slope on a graph using $m = \frac{rise}{run}$ .

It doesn’t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.

Example 9

Use the slope formula to find the slope of the line through the points $open parenthesis −2, −3 close parenthesis$ and $open parenthesis −7, 4 close parenthesis$ .

Show solution

**Table 6.15**
We'll call $open parenthesis −2, −3 close parenthesis$ point #1 and $open parenthesis −7, 4 close parenthesis$ point #2.	$open parenthesis −2, −3 x sub 1, y sub 1 close parenthesis open parenthesis −7, 4 x sub 2, y sub 2 close parenthesis$
Use the slope formula.	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .
Substitute the values.
$y$ of the second point minus $y$ of the first point	$m = \frac{4 - (−3)}{x_{2} - x_{1}}$ .
$x$ of the second point minus $x$ of the first point	$m = \frac{4 - (−3)}{−7 - (−2)}$ .
Simplify.	$\begin{matrix} m = \frac{7}{−5} \\ m = - \frac{7}{5} \end{matrix}$

Let’s verify this slope on the graph shown.

$\begin{matrix} m & = & \frac{rise}{run} \\ m & = & \frac{−7}{5} \\ m & = & - \frac{7}{5} \end{matrix}$

Practice Makes Perfect

Use Geoboards to Model Slope

In the following exercises, find the slope modeled on each geoboard.

An empty 5-by-5 geoboard grid with no rubber band placed yet. Users can drag endpoints to stretch a band between any two pegs, and a readout displays the resulting slope. Viewing window: x from -6.44 to 10.44, y from -3.22 to 7.22. — This interactive geoboard lets you explore slope by stretching rubber bands between pegs. Drag the endpoints to create lines with different rises and runs, and watch the slope value update in real time. Use this space to test your understanding of how vertical and horizontal changes affect the steepness and direction of a line.

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 2, forming a line.

Show solution

$\frac{1}{4}$

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 4 and the peg in column 5, row 2, forming a line.

Show solution

$\frac{2}{3}$

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 3, row 4 and the peg in column 5, row 1, forming a line.

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 1 and the peg in column 4, row 4, forming a line.

Show solution

$\frac{−3}{2} = - \frac{3}{2}$

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 5, row 4, forming a line.

Show solution

$- \frac{3}{4}$

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 2 and the peg in column 4, row 5, forming a line.

In the following exercises, model each slope. Draw a picture to show your results.

$\frac{2}{3}$

Show solution

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 5 and the peg in column 5, row 3, forming a line.

$\frac{3}{4}$

$\frac{1}{4}$

Show solution

$\frac{4}{3}$

$- \frac{1}{2}$

Show solution

$- \frac{3}{4}$

$- \frac{2}{3}$

Show solution

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2 and the peg in column 4, row 4, forming a line.

$- \frac{3}{2}$

Use $m = \frac{rise}{run}$ to find the Slope of a Line from its Graph

In the following exercises, find the slope of each line shown.

The line rises through (0, -4), (5, -2), and (10, 0). Labeled ticks run from -2 to 10 on the x-axis and -10 to 2 on the y-axis.

Show solution

$\frac{2}{5}$

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line passes through the points (negative 2, negative 8) and (2, negative 2).

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line passes through the points (negative 4, negative 6) and (4, 4).

Show solution

$\frac{5}{4}$

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line intercepts the y-axis at (0, negative 2) and passes through the point (3, 3).

The graph shows the x y coordinate plane. Labeled ticks run from -8 to 8 on both axes. A line passes through the points (negative 3, 3) and (3, 1).

Show solution

$- \frac{1}{3}$

The graph shows the x y coordinate plane. Labeled ticks run from -8 to 8 on both axes. A line passes through the points (negative 2, 4) and (2, 2).

The graph shows the x y coordinate plane. Labeled ticks run from -8 to 8 on both axes. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).

Show solution

$- \frac{3}{4}$

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line passes through the point (negative 3, 1) and intercepts the y-axis at (0, negative 1).

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line passes through the points (negative 2, 1) and (2, 4).

Show solution

$\frac{3}{4}$

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line passes through the points (negative 1, 1) and (2, 3).

A descending line passes through the explicitly marked points (-1, 6) and (1, 1). Both coordinate axes use equally spaced unit ticks; x is horizontal and y is vertical. The displayed line is the precise editorial graph for this exercise.

Show solution

$- \frac{5}{2}$

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line passes through the point (negative 1, 3) and intercepts the x-axis at (3, 0).

The graph shows the x y coordinate plane. Labeled ticks run from -8 to 8 on both axes. A line passes through the points (negative 2, 6) and (1, 4).

Show solution

$- \frac{2}{3}$

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line passes through the points (negative 1, 3) and (1, 2).

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).

Show solution

$\frac{1}{4}$

The graph shows the x y coordinate plane. Labeled ticks run from -6 to 6 on both axes. A line passes through the points (4, 2) and (7, 3).

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

$open parenthesis 1, 4 close parenthesis, open parenthesis 3, 9 close parenthesis$

Show solution

$\frac{5}{2}$

$open parenthesis 2, 3 close parenthesis, open parenthesis 5, 7 close parenthesis$

$open parenthesis 0, 3 close parenthesis, open parenthesis 4, 6 close parenthesis$

Show solution

$\frac{3}{4}$

$open parenthesis 0, 1 close parenthesis, open parenthesis 5, 4 close parenthesis$

$open parenthesis 2, 5 close parenthesis, open parenthesis 4, 0 close parenthesis$

Show solution

$- \frac{5}{2}$

$open parenthesis 3, 6 close parenthesis, open parenthesis 8, 0 close parenthesis$

$open parenthesis −3, 3 close parenthesis, open parenthesis 4, −5 close parenthesis$

Show solution

$- \frac{8}{7}$

$open parenthesis −2, 4 close parenthesis, open parenthesis 3, −1 close parenthesis$

$open parenthesis −1, −2 close parenthesis, open parenthesis 2, 5 close parenthesis$

Show solution

$\frac{7}{3}$

$open parenthesis −2, −1 close parenthesis, open parenthesis 6, 5 close parenthesis$

$open parenthesis 4, −5 close parenthesis, open parenthesis 1, −2 close parenthesis$

Show solution

$−1$

$open parenthesis 3, −6 close parenthesis, open parenthesis 2, −2 close parenthesis$

Corequisite Skills for Introductory Statistics is a custom course edition compiled for Introductory Statistics corequisite courses from OpenStax Contemporary Mathematics, Elementary Algebra 2e, Intermediate Algebra 2e, Prealgebra 2e. Content by OpenStax is reproduced as published, apart from any corrections recorded for this edition. Sections are renumbered to follow the course outline. The course outline is adapted from Larry Green's Support Course for Elementary Statistics (LibreTexts, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Support_Course_for_Elementary_Statistics), used under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Lessons 2.4 and 6.7 are written for this edition by XYZ Homework. Every OpenStax lesson is abridged to the subsections and practice exercises this course uses. Lessons are headed with their course titles, not the source section titles. Access the complete originals free at openstax.org. License: CC-BY-NC-SA-4.0.

6.4 Slope of a Line

Slope of a Line

Use Geoboards to Model Slope

Use m=riserun to Find the Slope of a Line from its Graph

Use the Slope Formula to find the Slope of a Line Between Two Points

Practice Makes Perfect

Use $m = \frac{rise}{run}$ to Find the Slope of a Line from its Graph