Scott and Brooke are organizing a fundraiser for their school. They are planning a pasta dinner, where adult tickets will cost $16 and kids' tickets will cost $8. Their goal is to make $2000. If they sell only adult tickets, how many must they sell to reach their goal? If they sell only kids' tickets, how many must they sell to reach their goal?
Guidance
When a line is in standard form, there are two different ways to graph it. The first is to change the equation to slope-intercept form and then graph as shown in the previous concept. The second is to use standard form to find the \begin{align*}x\end{align*} and @$\begin{align*}y-\end{align*}@$ intercepts of the line and connect the two. Here are a few examples.
Example A
Graph @$\begin{align*}5x - 2y = -15\end{align*}@$ .
Solution: Let’s use approach #1; change the equation to slope-intercept form.
@$$\begin{align*}5x -2y &= -15\\ -2y &= -5x - 15\\ y &= \frac{5}{2}x + \frac{15}{2}\end{align*}@$$
The @$\begin{align*}y-\end{align*}@$ intercept is @$\begin{align*}\left( 0, \frac{15}{2} \right)\end{align*}@$ . Change the improper fraction to a decimal and approximate it on the graph, (0, 7.5). Then use slope triangles. If you find yourself running out of room “rising 5” and “running 2,” you could also “fall 5” and “run backwards 2” to find a point on the other side of the @$\begin{align*}y-\end{align*}@$ intercept.
Example B
Find the x and y intercepts of the equation @$\begin{align*}4x - 3x = 21\end{align*}@$ .
Solution: Recall the Standard Form concept. The other coordinate will be zero at these points. Therefore, for the @$\begin{align*}x-\end{align*}@$ intercept, plug in zero for @$\begin{align*}y\end{align*}@$ and for the @$\begin{align*}y-\end{align*}@$ intercept, plug in zero for @$\begin{align*}x\end{align*}@$ .
@$$\begin{align*}4x - 3(0) &= 21 && 4(0) -3y = 21\\ 4x &= 21 && \quad \ \ -3y = 21\\ x &= \frac{21}{4} \ or \ 5.25 && \qquad \quad \ y = -7\end{align*}@$$
Example C
Graph the equation from Example B .
Solution: Use approach #2 from above. Plot each intercept from Example B on their respective axes and draw a line to connect them.
Intro Problem Revisit The equation, in standard form, for the pasta dinner sales goal is @$\begin{align*}2000 = 16x + 8y\end{align*}@$ . If they sell only adult tickets, we are looking for the x -intercept, so set y equal to zero.
@$$\begin{align*}2000 = 16x + 8(0)\\ 2000 = 16x\\ x = 125\end{align*}@$$
Therefore, they must sell 125 adult tickets to reach their goal.
If they sell only kids' tickets, we are looking for the y -intercept, so set x equal to zero.
@$$\begin{align*}2000 = 16(0) + 8y\\ 2000 = 8y\\ y = 250\end{align*}@$$
Therefore, they must sell 250 kids' tickets to reach their goal.
Guided Practice
1. Graph @$\begin{align*}4x + 6y = 18\end{align*}@$ by changing it into slope-intercept form.
2. Graph @$\begin{align*}5x - 3y = 30\end{align*}@$ by plotting the intercepts.
Answers
1. Change @$\begin{align*}4x + 6y = 18\end{align*}@$ into slope-intercept form by solving for @$\begin{align*}y\end{align*}@$ , then graph.
@$$\begin{align*}4x +6y &= 18\\ 6y &= -4x + 18\\ y &= - \frac{2}{3}x + 3\end{align*}@$$
2. Substitute in zero for @$\begin{align*}x\end{align*}@$ , followed by @$\begin{align*}y\end{align*}@$ and solve each equation.
@$$\begin{align*}5(0) - 3y &= 30 && 5x -3(0) = 30\\ -3y &= 30 && \qquad \quad 5x = 30\\ y &= -10 && \qquad \quad \ x = 6\end{align*}@$$
Now, plot each on their respective axes and draw a line.
Explore More
Graph the following lines by changing the equation to slope-intercept form.
- @$\begin{align*}-2x + y = 5\end{align*}@$
- @$\begin{align*}3x + 8y = 16\end{align*}@$
- @$\begin{align*}4x -2y = 10\end{align*}@$
- @$\begin{align*}6x + 5y = -20\end{align*}@$
- @$\begin{align*}9x - 6y = 24\end{align*}@$
- @$\begin{align*}x + 4y = -12\end{align*}@$
Graph the following lines by finding the intercepts.
- @$\begin{align*}2x + 3y = 12\end{align*}@$
- @$\begin{align*}-4x + 5y = 30\end{align*}@$
- @$\begin{align*}x - 2y = 8\end{align*}@$
- @$\begin{align*}7x + y = -7\end{align*}@$
- @$\begin{align*}6x + 10y = 15\end{align*}@$
- @$\begin{align*}4x -8y = -28\end{align*}@$
- @$\begin{align*}y=3\end{align*}@$
- Writing Which method do you think is easier? Why?
- Writing Which method would you use to graph @$\begin{align*}x = -5\end{align*}@$ ? Why?