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In the previous example, the three points we found were easy to graph. But this is not always the case. Let’s see what happens in the equation $2x+y=3.$ If $y$ is $0,$ what is the value of $x?$
The solution is the point $\left(\frac{3}{2},0\right).$ This point has a fraction for the $x$ -coordinate. While we could graph this point, it is hard to be precise graphing fractions. Remember in the example $y=\frac{1}{2}x+3,$ we carefully chose values for $x$ so as not to graph fractions at all. If we solve the equation $2x+y=3$ for $y,$ it will be easier to find three solutions to the equation.
Now we can choose values for $x$ that will give coordinates that are integers. The solutions for $x=0,x=1,$ and $x=\mathrm{-1}$ are shown.
$y=\mathrm{-2}x+3$ | ||
---|---|---|
$x$ | $y$ | $(x,y)$ |
$0$ | $5$ | $(\mathrm{-1},5)$ |
$1$ | $3$ | $(0,3)$ |
$\mathrm{-1}$ | $1$ | $(1,1)$ |
Graph the equation $3x+y=\mathrm{-1}.$
Find three points that are solutions to the equation.
First, solve the equation for $y.$
$\begin{array}{ccc}\hfill 3x+y& =& \mathrm{-1}\hfill \\ \hfill y& =& \mathrm{-3}x-1\hfill \end{array}$
We’ll let $x$ be $0,1,$ and $\mathrm{-1}$ to find three points. The ordered pairs are shown in the table. Plot the points, check that they line up, and draw the line.
$y=\mathrm{-3}x-1$ | ||
---|---|---|
$x$ | $y$ | $(x,y)$ |
$0$ | $\mathrm{-1}$ | $(0,\mathrm{-1})$ |
$1$ | $\mathrm{-4}$ | $(1,\mathrm{-4})$ |
$\mathrm{-1}$ | $2$ | $(\mathrm{-1},2)$ |
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 $x\text{-}$ and $y$ -axes are the same, the graphs match.
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 $x=\mathrm{-3}.$ 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 $\mathrm{-3}.$
To make a table of solutions, we write $\mathrm{-3}$ for all the $x$ values. Then choose any values for $y.$ Since $x$ does not depend on $y,$ you can chose any numbers you like. But to fit the size of our coordinate graph, we’ll use $1,2,$ and $3$ for the $y$ -coordinates as shown in the table.
$x=\mathrm{-3}$ | ||
---|---|---|
$x$ | $y$ | $(x,y)$ |
$\mathrm{-3}$ | $1$ | $(\mathrm{-3},1)$ |
$\mathrm{-3}$ | $2$ | $(\mathrm{-3},2)$ |
$\mathrm{-3}$ | $3$ | $(\mathrm{-3},3)$ |
Then plot the points and connect them with a straight line. Notice in [link] that the graph is a vertical line .
A vertical line is the graph of an equation that can be written in the form $x=a.$
The line passes through the $x$ -axis at $\left(a,0\right)$ .
Graph the equation $x=2.$ What type of line does it form?
The equation has only variable, $x,$ and $x$ is always equal to $2.$ We make a table where $x$ is always $2$ and we put in any values for $y.$
$x=2$ | ||
---|---|---|
$x$ | $y$ | $(x,y)$ |
$2$ | $1$ | $(2,1)$ |
$2$ | $2$ | $(2,2)$ |
$2$ | $3$ | $(2,3)$ |
Plot the points and connect them as shown.
The graph is a vertical line passing through the $x$ -axis at $2.$
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 $x.$
To make a table of solutions, write $4$ for all the $y$ values and then choose any values for $x.$
We’ll use $0,2,$ and $4$ for the $x$ -values.
$y=4$ | ||
---|---|---|
$x$ | $y$ | $(x,y)$ |
$0$ | $4$ | $(0,4)$ |
$2$ | $4$ | $(2,4)$ |
$4$ | $4$ | $(4,4)$ |
Plot the points and connect them, as shown in [link] . This graph is a horizontal line passing through the $y\text{-axis}$ at $4.$
A horizontal line is the graph of an equation that can be written in the form $y=b.$
The line passes through the $y\text{-axis}$ at $\left(0,b\right).$
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