The three linear equations discussed so far each provide information about the line. The following table summarizes the information each form provides:
| Form Name | Information | Example |
|---|---|---|
| Slope Intercept | Provides the slope \(m\) and the \(y\)-intercept \((0,b)\) |
\(y=\large{\frac{4}{3}}\normalsize{x-6}\) slope = \(\large{\frac{4}{3}}\) \(y\)-intercept = \((0,-6)\) |
| Point Slope | Provides the slope \(m\) and a point \((x_p,y_p)\) |
\(y-5=-\large{\frac{2}{7}}\normalsize{(x+9)}\) slope = \(-\large{\frac{2}{7}}\) point = \((-9,5)\) |
| Standard | Find intercepts \(x\)-intercept put \(y=0\) \(y\)-intercept put \(x=0\) |
\(-6x+4y=36\) \(x\)-intercept \((-6,0)\) \(y\)-intercept \((0,9)\) |
Since the slope intercept form is the unique equation of a line, it makes sense to convert from standard, or point slope, into slope intercept. The keys are outlined below and the following video shows some examples.
The last portion is about being able to decipher from a word problem which form is the best to create an equation. Being able to create an equation to help interpret the context of the problem and answer specific questions about the problem are the key ideas.
Bob is taking a road trip. His car has a full tank of gas and holds \(16\) gallons. While traveling, Bob notices he is using \(2\) gallons per hour. Write an equation that models Bob's trip. If the next rest stop is \(2.5\) hours away, how many gallons will Bob still have in his car?
Let \(x\) represent the time travelled and let \(y\) represent the gallons of gas. Since Bob's car starts with a full tank of gas, the \(16\) gallons represents the \(y\)-intercept. The car is using \(2\) gallons every hour, which is the slope of the line. The slope is negative here because the car is using \(2\) gallons every hour.
Therefore, the slope intercept form makes the best sense here so the equation is: \(y=16-2x\).
The \(2.5\) hours is a time input, so \(x=2.5\), and substituting into our model equation yields:
$$\begin{array}{l} y=16-2(2.5) \\ \implies y=16-5 \\ \implies y=11\;\color{green}{\checkmark} \end{array}$$
Bob will have \(11\) gallons of gas left in his car after \(2.5\) hours of driving.
Your family is on a ski vacation. Lift tickets cost \(\$80\) per day. Snowboard rentals cost \(\$40\) per day. Your family spent \(\$480\) on lift tickets and snowboard rentals. Write an equation that models this problem. What is the meaning of the \(x\) and \(y\)-intercepts in the context of the problem?
Let \(x\) represent the number of lift tickets and \(y\) be the number of snowboard rentals. Since there is a cost for each item, and no slope mentioned, the standard form makes the best sense. Multiplying the cost per day with how many days on the lift tickets, and the cost per day with how many snowboard rentals, the equation becomes: \(80x+40y=480\).
To find the intercepts, plug in \(0\) for the appropriate variable:
$$\begin{array}{l} x-\text{intercept} & y=0 & 80x+40(0)=480 \\ & & \implies 80x=480 \\ & & \implies x=\frac{480}{80} \\ & & \implies x=6\;\color{green}{\checkmark} \\ y-\text{intercept} & x=0 & 80(0)+40y=480 \\ & & \implies 40y=480 \\ & & \implies y=\frac{480}{40} \\ & & \implies y=12\;\color{green}{\checkmark}\end{array}$$
The \(x\)-intercept of \((6,0)\) means you purchased \(6\) days of lift tickets and rented no snowboards. The \(y\)-intercept of \((0,12)\) means you rented \(12\) snowboards but purchased no lift tickets.
Biologists have found that the number of chirps some crickets make per minute is related to temperature. When crickets chirp \(124\) times a minute, it is about \(68^\circ\) Fahrenheit. When they chirp \(172\) times a minute, it is about \(80^\circ\) Fahrenheit. Write an equation to model this problem. How warm is it when the crickets are chirping \(150\) times a minute?
Let \(x\) represent the number of chirps and \(y\) denote the temperature. This problem poses information that sets up as two points: \((124,68)\) and \((172,80)\). The first thing to do is find the slope:
$$m=\frac{80-68}{172-124}\implies m=\frac{12}{48}\implies m=\frac{1}{4}$$
The interpretation of the slope is the temperature goes up \(1^\circ\) for every increase of \(4\) chirps.
To find an equation, use either point and the slope to create a point slope equation: \(y-80=\large{\frac{1}{4}}\normalsize{(x-172)}\).
When the crickets chirp \(150\) times a minute, plug in \(x=150\), to the model equation:
$$\begin{array}{l} y-80=\frac{1}{4}(150-172) \\ \implies y-80=\frac{1}{4}(-22) \\ \implies y-80=-5.5 \\ \implies y=-5.5+80 \\ \implies y=74.5\;\color{green}{\checkmark} \end{array}$$
The temperature is about \(74.5^\circ\) Fahrenheit when the crickets chirp 150 times per minute.