The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 3 x is plotted as an arrow extending from the top left toward the bottom right.

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$y = \frac{1}{2} x + 3$

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$x - y = 6$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus y equals 6 is plotted as an arrow extending from the bottom left toward the top right.

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$2 x + y = 7$

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$3 x - 2 y = 6$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 3 x minus 2 y equals 6 is plotted as an arrow extending from the bottom left toward the top right.

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Graph Vertical and Horizontal lines

In the following exercises, graph each equation.

$y = −2$

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$x = 3$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x equals 3 is plotted as a vertical line.

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In the following exercises, graph each pair of equations in the same rectangular coordinate system.

$y = −2 x$ and $y = −2$

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$y = \frac{4}{3} x$ and $y = \frac{4}{3}$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals four-thirds x is plotted as an arrow extending from the bottom left toward the top right. The line y equals four-thirds is plotted as a horizontal line.

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Graphing with Intercepts

Identify the x - and y -Intercepts on a Graph

In the following exercises, find the x - and y -intercepts.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.

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The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (3, 0) and (0, 3) is plotted.

$(3, 0), (0, 3)$

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Find the x - and y -Intercepts from an Equation of a Line

In the following exercises, find the intercepts of each equation.

$x + y = 5$

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$x - y = −1$

$(−1, 0), (0, 1)$

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$x + 2 y = 6$

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$2 x + 3 y = 12$

$(6, 0), (0, 4)$

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$y = \frac{3}{4} x - 12$

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$y = 3 x$

$(0, 0)$

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Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

$− x + 3 y = 3$

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$x + y = −2$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 2 is plotted as an arrow extending from the top left toward the bottom right.

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$x - y = 4$

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$2 x - y = 5$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 2 x minus y equals 5 is plotted as an arrow extending from the bottom left toward the top right.

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$2 x - 4 y = 8$

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$y = 2 x$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 x is plotted as an arrow extending from the bottom left toward the top right.

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Slope of a Line

Use Geoboards to Model Slope

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

The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.

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The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.

$\frac{4}{3}$

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The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.

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The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.

$- \frac{2}{3}$

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

$\frac{1}{3}$

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$\frac{3}{2}$

The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 3 row 2.

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$- \frac{2}{3}$

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$- \frac{1}{2}$

The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 2 and the point in column 3 row 3.

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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 graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 1, 3), (0, 0), and (1, negative 3) is plotted.

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The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 6) and (5, 2) is plotted.

$- \frac{1}{2}$

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Find the Slope of Horizontal and Vertical Lines

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

$y = 2$

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$x = 5$

undefined

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$x = −3$

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$y = −1$

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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.

$(−1, −1), (0, 5)$

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$(3, 5), (4, −1)$

$−6$

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$(−5, −2), (3, 2)$

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$(2, 1), (4, 6)$

$\frac{5}{2}$

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Graph a Line Given a Point and the Slope

In the following exercises, graph each line with the given point and slope.

$(2, −2)$ ; $m = \frac{5}{2}$

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$(−3, 4)$ ; $m = - \frac{1}{3}$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 4) and (0, 3) is plotted.

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x -intercept $−4$ ; $m = 3$

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y -intercept 1; $m = - \frac{3}{4}$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (0, 1) and (4, negative 2) is plotted.

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Solve Slope Applications

In the following exercises, solve these slope applications.

The roof pictured below has a rise of 10 feet and a run of 15 feet. What is its slope?

The figure shows a person on a ladder using a hammer on the roof of a building.

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A mountain road rises 50 feet for a 500-foot run. What is its slope?

$\frac{1}{10}$

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Intercept Form of an Equation of a Line

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

In the following exercises, use the graph to find the slope and y -intercept of each line. Compare the values to the equation $y = m x + b$ .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 4 x minus 1 is plotted from the lower left to the top right.

$y = 4 x - 1$

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The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x plus 4 is plotted from the top left to the bottom right.

$y = - \frac{2}{3} x + 4$

slope $m = - \frac{2}{3}$ and y -intercept $(0, 4)$

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Identify the Slope and y-Intercept from an Equation of a Line

In the following exercises, identify the slope and y -intercept of each line.

$y = −4 x + 9$

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$y = \frac{5}{3} x - 6$

$\frac{5}{3}; (0, −6)$

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$5 x + y = 10$

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$4 x - 5 y = 8$

$\frac{4}{5}; (0, - \frac{8}{5})$

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Graph a Line Using Its Slope and Intercept

In the following exercises, graph the line of each equation using its slope and y -intercept.

$y = 2 x + 3$

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$y = − x - 1$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x minus 1 is plotted from the top left to the bottom right.

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$y = - \frac{2}{5} x + 3$

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$4 x - 3 y = 12$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 4 x minus 3 y equals 12 is plotted from the bottom left to the top right.

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In the following exercises, determine the most convenient method to graph each line.

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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