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Graph the equation 3 x + y = −1 .

Solution

Find three points that are solutions to the equation. 3 x + y = −1 First solve the equation for y . y = −3 x 1

We’ll let x be 0, 1, and −1 to find 3 points. The ordered pairs are shown in [link] . Plot the points, check that they line up, and draw the line. See [link] .

3 x + y = −1
x y ( x , y )
0 −1 ( 0 , −1 )
1 −4 ( 1 , −4 )
−1 2 ( −1 , 2 )
The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (negative 1, 2), (0, negative 1), and (1, negative 4). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation 3x plus y equals negative 1.
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Graph the equation 2 x + y = 2 .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 4, 10), (negative 2, 6), (0, 2), (2, negative 2), (4, negative 6), and (6, negative 10). The line has arrows on both ends pointing to the outside of the figure.

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Graph the equation 4 x + y = −3 .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 3, 9), (negative 2, 5), (negative 1, 1), (0, negative 3), (1, negative 7), and (2, negative 10). The line has arrows on both ends pointing to the outside of the figure.

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If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x - and y -axis are the same, the graphs match!

The equation in [link] was written in standard form, with both x and y on the same side. We solved that equation for y in just one step. But for other equations in standard form it is not that easy to solve for y , so we will leave them in standard form. We can still find a first point to plot by letting x = 0 and solving for y . We can plot a second point by letting y = 0 and then solving for x . Then we will plot a third point by using some other value for x or y .

Graph the equation 2 x 3 y = 6 .

Solution

Find three points that are solutions to the equation. 2 x 3 y = 6 First let x = 0 . 2 ( 0 ) 3 y = 6 Solve for y . −3 y = 6 y = −2 Now let y = 0 . 2 x 3 ( 0 ) = 6 Solve for x . 2 x = 6 x = 3 We need a third point. Remember, we can choose any value for x or y . We’ll let x = 6 . 2 ( 6 ) 3 y = 6 Solve for y . 12 3 y = 6 −3 y = −6 y = 2

We list the ordered pairs in [link] . Plot the points, check that they line up, and draw the line. See [link] .

2 x 3 y = 6
x y ( x , y )
0 −2 ( 0 , −2 )
3 0 ( 3 , 0 )
6 2 ( 6 , 2 )
The figure shows a straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (0, negative 2), (3, 0), and (6, 2). A straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation 2x minus 3y equals 6.
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Graph the equation 4 x + 2 y = 8 .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), and (4, negative 4). The line has arrows on both ends pointing to the outside of the figure.

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Graph the equation 2 x 4 y = 8 .

The figure shows a straight line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line goes through the points (negative 6, negative 5), (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), and (6, 1). The line has arrows on both ends pointing to the outside of the figure.

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Graph vertical and horizontal lines

Can we graph an equation with only one variable? Just x and no y , or just y without an x ? How will we make a table of values to get the points to plot?

Let’s consider the equation x = −3 . This equation has only one variable, x . The equation says that x is always equal to −3 , so its value does not depend on y . No matter what y is, the value of x is always −3 .

So to make a table of values, write −3 in for all the x values. Then choose any values for y . Since x does not depend on y , you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y -coordinates. See [link] .

x = −3
x y ( x , y )
−3 1 ( −3 , 1 )
−3 2 ( −3 , 2 )
−3 3 ( −3 , 3 )

Plot the points from [link] and connect them with a straight line. Notice in [link] that we have graphed a vertical line .

The figure shows a vertical straight line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (negative 3, 1), (negative 3, 2), and (negative 3, 3). A vertical straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation x equals negative 3.

Vertical line

A vertical line    is the graph of an equation of the form x = a .

The line passes through the x -axis at ( a , 0 ) .

 

Graph the equation x = 2 .

Solution

The equation has only one variable, x , and x is always equal to 2. We create [link] where x is always 2 and then put in any values for y . The graph is a vertical line passing through the x -axis at 2. See [link] .

x = 2
x y ( x , y )
2 1 ( 2 , 1 )
2 2 ( 2 , 2 )
2 3 ( 2 , 3 )
The figure shows a straight vertical line drawn through three points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the three points which are labeled by their ordered pairs (2, 1), (2, 2), and (2, 3). A vertical straight line goes through all three points. The line has arrows on both ends pointing to the outside of the figure. The line is labeled with the equation x equals 2.
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Graph the equation x = 5 .

The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (5, 1), (5, 2), (5, 3), and all other points with first coordinate 5. The line has arrows on both ends pointing to the outside of the figure.

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Graph the equation x = −2 .

The figure shows a straight vertical line drawn on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The straight line goes through the points (negative 2, 1), (negative 2, 2), (negative 2, 3), and all other points with first coordinate negative 2. The line has arrows on both ends pointing to the outside of the figure.

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What if the equation has y but no x ? Let’s graph the equation y = 4 . This time the y - value is a constant, so in this equation, y does not depend on x . Fill in 4 for all the y ’s in [link] and then choose any values for x . We’ll use 0, 2, and 4 for the x -coordinates.

y = 4
x y ( x , y )
0 4 ( 0 , 4 )
2 4 ( 2 , 4 )
4 4 ( 4 , 4 )

The graph is a horizontal line passing through the y -axis at 4. See [link] .

Practice Key Terms 3

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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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