# 4.2 Graph linear equations in two variables  (Page 3/7)

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Graph the equation $3x+y=-1$ .

## Solution

$\begin{array}{cccccc}\text{Find three points that are solutions to the equation.}\hfill & & & \hfill 3x+y& =\hfill & -1\hfill \\ \text{First solve the equation for}\phantom{\rule{0.2em}{0ex}}y.\hfill & & & \hfill y& =\hfill & -3x-1\hfill \end{array}$

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] .

 $3x+y=-1$ $x$ $y$ $\left(x,y\right)$ 0 $-1$ $\left(0,-1\right)$ 1 $-4$ $\left(1,-4\right)$ $-1$ 2 $\left(-1,2\right)$

Graph the equation $2x+y=2$ .

Graph the equation $4x+y=-3$ .

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 $2x-3y=6$ .

## Solution

$\begin{array}{cccccc}\begin{array}{c}\text{Find three points that are solutions to the}\hfill \\ \text{equation.}\hfill \end{array}\hfill & & & \hfill 2x-3y& =\hfill & 6\hfill \\ \text{First let}\phantom{\rule{0.2em}{0ex}}x=0.\hfill & & & \hfill 2\left(0\right)-3y& =\hfill & 6\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}y.\hfill & & & \hfill -3y& =\hfill & 6\hfill \\ & & & \hfill y& =\hfill & -2\hfill \\ \\ \\ \text{Now let}\phantom{\rule{0.2em}{0ex}}y=0.\hfill & & & \hfill 2x-3\left(0\right)& =\hfill & 6\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}x.\hfill & & & \hfill 2x& =\hfill & 6\hfill \\ & & & \hfill x& =\hfill & 3\hfill \\ \begin{array}{c}\text{We need a third point. Remember, we can}\hfill \\ \text{choose any value for}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}y.\phantom{\rule{0.2em}{0ex}}\text{We’ll let}\phantom{\rule{0.2em}{0ex}}x=6.\hfill \end{array}\hfill & & & \hfill 2\left(6\right)-3y& =\hfill & 6\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}y.\hfill & & & \hfill 12-3y& =\hfill & 6\hfill \\ & & & \hfill -3y& =\hfill & -6\hfill \\ & & & \hfill y& =\hfill & 2\hfill \end{array}$

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 $-2$ $\left(0,-2\right)$ 3 0 $\left(3,0\right)$ 6 2 $\left(6,2\right)$

Graph the equation $4x+2y=8$ .

Graph the equation $2x-4y=8$ .

## 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] .

 $x=-3$ $x$ $y$ $\left(x,y\right)$ $-3$ 1 $\left(-3,1\right)$ $-3$ 2 $\left(-3,2\right)$ $-3$ 3 $\left(-3,3\right)$

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

## 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 $x=2$ .

## 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] .

 $x=2$ $x$ $y$ $\left(x,y\right)$ 2 1 $\left(2,1\right)$ 2 2 $\left(2,2\right)$ 2 3 $\left(2,3\right)$

Graph the equation $x=5$ .

Graph the equation $x=-2$ .

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 $x$ . 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.

 $y=4$ $x$ $y$ $\left(x,y\right)$ 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] .

how did you get the value of 2000N.What calculations are needed to arrive at it
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Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
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Tremayne
x-3y =1; 3x-2y+4=0 graph
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