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Graph the equation $3x+y=\mathrm{-1}$ .
$\begin{array}{cccccc}\text{Find three points that are solutions to the equation.}\hfill & & & \hfill 3x+y& =\hfill & \mathrm{-1}\hfill \\ \text{First solve the equation for}\phantom{\rule{0.2em}{0ex}}y.\hfill & & & \hfill y& =\hfill & \mathrm{-3}x-1\hfill \end{array}$
We’ll let $x$ be 0, 1, and $\mathrm{-1}$ 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] .
$3x+y=\mathrm{-1}$ | ||
$x$ | $y$ | $\left(x,y\right)$ |
0 | $\mathrm{-1}$ | $\left(0,\mathrm{-1}\right)$ |
1 | $\mathrm{-4}$ | $\left(1,\mathrm{-4}\right)$ |
$\mathrm{-1}$ | 2 | $\left(\mathrm{-1},2\right)$ |
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 $2x-3y=6$ .
$\begin{array}{cccccc}\begin{array}{c}\text{Find three points that are solutions to the}\hfill \\ \text{equation.}\hfill \end{array}\hfill & & & \hfill 2x-3y& =\hfill & 6\hfill \\ \text{First let}\phantom{\rule{0.2em}{0ex}}x=0.\hfill & & & \hfill 2\left(0\right)-3y& =\hfill & 6\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}y.\hfill & & & \hfill \mathrm{-3}y& =\hfill & 6\hfill \\ & & & \hfill y& =\hfill & \mathrm{-2}\hfill \\ \\ \\ \text{Now let}\phantom{\rule{0.2em}{0ex}}y=0.\hfill & & & \hfill 2x-3\left(0\right)& =\hfill & 6\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}x.\hfill & & & \hfill 2x& =\hfill & 6\hfill \\ & & & \hfill x& =\hfill & 3\hfill \\ \begin{array}{c}\text{We need a third point. Remember, we can}\hfill \\ \text{choose any value for}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}y.\phantom{\rule{0.2em}{0ex}}\text{We\u2019ll let}\phantom{\rule{0.2em}{0ex}}x=6.\hfill \end{array}\hfill & & & \hfill 2\left(6\right)-3y& =\hfill & 6\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}y.\hfill & & & \hfill 12-3y& =\hfill & 6\hfill \\ & & & \hfill \mathrm{-3}y& =\hfill & \mathrm{-6}\hfill \\ & & & \hfill y& =\hfill & 2\hfill \end{array}$
We list the ordered pairs in [link] . Plot the points, check that they line up, and draw the line. See [link] .
$2x-3y=6$ | ||
$x$ | $y$ | $\left(x,y\right)$ |
0 | $\mathrm{-2}$ | $\left(0,\mathrm{-2}\right)$ |
3 | 0 | $\left(3,0\right)$ |
6 | 2 | $\left(6,2\right)$ |
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 $x=\mathrm{-3}$ . This equation has only one variable, $x$ . The equation says that $x$ is always equal to $\mathrm{-3}$ , so its value does not depend on $y$ . No matter what $y$ is, the value of $x$ is always $\mathrm{-3}$ .
So to make a table of values, write $\mathrm{-3}$ 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] .
$x=\mathrm{-3}$ | ||
$x$ | $y$ | $\left(x,y\right)$ |
$\mathrm{-3}$ | 1 | $\left(\mathrm{-3},1\right)$ |
$\mathrm{-3}$ | 2 | $\left(\mathrm{-3},2\right)$ |
$\mathrm{-3}$ | 3 | $\left(\mathrm{-3},3\right)$ |
Plot the points from [link] and connect them with a straight line. Notice in [link] that we have graphed a vertical line .
A vertical line is the graph of an equation of the form $x=a$ .
The line passes through the x -axis at $\left(a,0\right)$ .
Graph the equation $x=2$ .
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 | $\left(2,1\right)$ |
2 | 2 | $\left(2,2\right)$ |
2 | 3 | $\left(2,3\right)$ |
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 | $\left(0,4\right)$ |
2 | 4 | $\left(2,4\right)$ |
4 | 4 | $\left(4,4\right)$ |
The graph is a horizontal line passing through the y -axis at 4. See [link] .
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