11.4 Understand slope of a line (Page 5/9)

Prealgebra

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Find the slope of the line through the given points: $(8, 5)$ and $(6, 3) .$

Find the slope of the line through the given points: $(1, 5)$ and $(5, 9) .$

How do we know which point to call #1 and which to call #2? Let’s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative.

We’ll call $(4, 5)$ point #1 and $(1, 2)$ point #2.	$\overset{x_{1}, y_{1}}{(4, 5)} and \overset{x_{2}, y_{2}}{(1, 2)}$
Use the slope formula.	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
Substitute the values in the slope formula:
$y$ of the second point minus $y$ of the first point	$m = \frac{2 - 5}{x_{2} - x_{1}}$
$x$ of the second point minus $x$ of the first point	$m = \frac{2 - 5}{1 - 4}$
Simplify the numerator and the denominator.	$m = \frac{−3}{−3}$
	$m = 1$

The slope is the same no matter which order we use the points.

Find the slope of the line through the points $(−2, −3)$ and $(−7, 4) .$

Solution

We’ll call $(−2, −3)$ point #1 and $(−7, 4)$ point #2.	$\overset{x_{1}, y_{1}}{(−2, −3)} and \overset{x_{2}, y_{2}}{(−7, 4)}$
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.	$m = \frac{7}{−5}$
	$m = - \frac{7}{5}$

Let’s confirm this on the graph shown.

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

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Find the slope of the line through the pair of points: $(−3, 4)$ and $(2, −1) .$

−1

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Find the slope of the line through the pair of points: $(−2, 6)$ and $(−3, −4) .$

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Graph a line given a point and the slope

In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

Another method we can use to graph lines is the point-slope method . Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.

Graph the line passing through the point $(1, −1)$ whose slope is $m = \frac{3}{4} .$

Solution

Plot the given point, $(1, −1) .$
The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -3 to 4. A labeled point is drawn at “ordered pair 1, -1”.

Use the slope formula $m = \frac{rise}{run}$ to identify the rise and the run.

\begin{matrix} m = \frac{3}{4} \\ \frac{rise}{run} = \frac{3}{4} \\ rise = 3 \\ run = 4 \end{matrix}

Starting at the point we plotted, count out the rise and run to mark the second point. We count $3$ units up and $4$ units right.
The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two line segments are drawn. A vertical line segment connects the points “ordered pair 1, -1” and “order pair “1, 2”. It is labeled “3”. A horizontal line segment starts at the top of the vertical line segment and goes to the right, connecting the points “ordered pair 1, 2” and “ordered pair 5, 2”. It is labeled “4”.

Then we connect the points with a line and draw arrows at the ends to show it continues.

We can check our line by starting at any point and counting up $3$ and to the right $4 .$ We should get to another point on the line.

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Graph the line passing through the point with the given slope:

$(2, −2), m = \frac{4}{3}$

The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -2, 3” and “ordered pair 8, 6”.

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Graph the line passing through the point with the given slope:

$(−2, 3), m = \frac{1}{4}$

The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -2, 3” and “ordered pair 2, 4”.

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Graph a line given a point and a slope.

Plot the given point.
Use the slope formula to identify the rise and the run.
Starting at the given point, count out the rise and run to mark the second point.
Connect the points with a line.

Graph the line with $y$ -intercept $(0, 2)$ and slope $m = - \frac{2}{3} .$

Solution

Plot the given point, the $y$ -intercept $(0, 2) .$
The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -1 to 3. The point “ordered pair 0, 2” is labeled.

Use the slope formula $m = \frac{rise}{run}$ to identify the rise and the run.

\begin{matrix} m = - \frac{2}{3} \\ \frac{rise}{run} = \frac{−2}{3} \\ rise = –2 \\ run = 3 \end{matrix}

Starting at $(0, 2),$ count the rise and the run and mark the second point.
The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line segment connects points at “ordered pair 0, 2” and “ordered pair 0, 0” and is labeled “down 2”. A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 0, 3” and is labeled “right 3”.

Connect the points with a line.
The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at “ordered pair 0, 2” and “ordered pair 3, 0”. A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 0, 2” and “ordered pair 0, 0 ”. A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 3, 0”.

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Graph the line with the given intercept and slope:

$y$ -intercept $4, m = - \frac{5}{2}$

The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, -6”.

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Graph the line with the given intercept and slope:

$x$ -intercept $−3, m = - \frac{3}{4}$

The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair -3, 0” and “ordered pair 8, -8”.

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Graph the line passing through the point $(−1, −3)$ whose slope is $m = 4 .$

Solution

Plot the given point.
The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The point “ordered pair -1, -3” is labeled.

Identify the rise and the run.	$m = 4$
Write 4 as a fraction.	$\frac{rise}{run} = \frac{4}{1}$
	$rise = 4 run = 1$

Count the rise and run.
The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The y-axis runs from -4 to 2. A vertical line segment connects points at “ordered pair -1, -3” and “ordered pair -1, 1” and is labeled “up 4”. A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1” and is labeled “over 1”.

Mark the second point. Connect the two points with a line.

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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