Back in Chapter 2, we’ve mostly been learning about evaluating functions: given a particular value of the independent variable, figuring out the resulting value of the dependent variable. You might call this process “plugging in.”
Here in Chapter 3, we will instead be interested in solving equations: given a particular value of the dependent variable, figuring out what value of the independent variable it came from.
When we are solving an equation, this means we know a value of the dependent variable (an output) and we want to figure out the corresponding value of the independent variable (the input) that produced it.
When we are evaluating a function, this means we know a value of the independent variable (an input) and we want to figure out the corresponding value of the dependent variable (the output).
Another way to think about the difference between evaluating and solving is that it is the difference between doing and undoing. When we evaluate a function, we are doing the steps in the equation to the input, so that we can figure out the corresponding output. On the other hand, when we solve an equation, we already have the output, so we want to undo all the things that the equation said to do. That way we can get back to the original input.
Oh no! Bleepblorp absolutely hates wearing shoes and socks! Write Bleepblorp a set of careful, step-by-step instructions about how to remove the footwear.
Your kitchen sink keeps getting clogged. Very annoying. Last time the plumber was able to fix it pretty quickly. But now the sink is clogged again. This time when the plumber comes and unclogs the sink, he suggests redoing the trap and a few other things that were causing the problem. You are pretty tired of it clogging up and tell him to “go ahead.” While you’re glad that the sink works when he’s done, you’re a bit shocked when his bill arrives a few days later for parts plus $278.75 in labor. Does that seem right?
Since $278.75 is between $250.00 and $325.00, we see that the time must be between 2 and 3 hours. You remember the plumber being there over 2 hours, so this is certainly a reasonable answer. Well, a lot of money, but mathematically it makes sense.
Let’s revisit the truck hauling bags of grass seed. Remember, the truck weighs 3,900 pounds when it is empty. Each bag of grass seed it carries weighs 4.2 pounds. The equation for the total weight is
\begin{equation*}
W = 3{,}900 + 4.2B
\end{equation*}
Still curious, you’d like to know exactly how many hours and minutes the plumber worked. We could use successive approximation. For example, for 2.5 hours the bill would have been
The plumber took about 2 hours, 23 minutes. Thinking back, the plumber had arrived around 10:30 in the morning and stayed past lunch, probably until around 1:00 p.m. That’s about right.
The equation for the total weight of the truck carrying bags of grass seed is
\begin{equation*}
W = 3{,}900 + 4.2B
\end{equation*}
Use successive approximation to find approximately how many bags the truck is carrying if its total weight is 14,500 pounds. Give your answer correct to the nearest bag (there’s no such thing as a decimal number of bags, anyway).
Wait a minute! We could have figured the plumber’s cost out much more quickly. If the labor cost was $278.75, we know the first $100 was the trip charge. That leaves
That worked well. But, can we figure out this sort of calculation in other problems? What is the general method we’re using? Can we write down our method in an organized fashion so that someone else could follow our thinking here? Turns out there is a formal way to show this calculation, called (symbolically) solving the equation. Officially any method of getting a solution to an equation is considered solving the equation, but in the rest of this book, and in most places that use algebra, when we refer to “solving the equation” or give the instruction to solve, we mean symbolically.
Here’s how it works. We want to figure out when \(P=278.75\text{.}\) We know from our equation that \(P = 100 + 75T\text{.}\) So we want to find the time \(T\) where
Remember that the equal sign means that the two sides are the same number. On the left-hand side we have \(100 + 75T\text{.}\) On the right-hand side we have \(278.75\text{.}\) Looks different, but same thing: both of them are \(P\text{.}\) We are looking for the value of \(T\) that makes these two sides equal.
The first thing we did was subtract the $100 trip charge. In this formal method, we subtract 100 from each side of our equation. If the left-hand side and the right-hand side are the same number, then they will still be equal when we take away 100 from each side. We get
The next thing we did to figure out the answer was divide by the $75/hour charge. In this formal method, we can divide each side of our equation by 75. Again, if the left-hand side and right-hand side are the same number, then they will still be equal when we divide by 75. Here goes.
Notice that we wrote the division in fraction form (instead of using \(\div\)). To understand why the 75’s cancelled, remember that \(75T\) is short for \(75\ast T\) and so
This calculation is a good example of the Shoes and Socks Principle. To evaluate the function \(P = 100 + 75T\text{,}\) we would do the following steps:
And so, when we solved, we did the opposite steps in the opposite order. The opposite of adding 100 is subtracting 100, and the opposite of multiplying by 75 is dividing by 75:
In previous activities in this chapter, you evaluated this equation at many different values of \(B\text{.}\) What steps do you do, in what order, to evaluate the equation?
Let’s practice working with this symbolic way of solving equations. Suppose instead the plumber went to my neighbor’s house and billed her for $160 in labor costs. How long did the plumber work at my neighbor’s? As before, we begin with our equation
\begin{equation*}
P = 100 + 75T
\end{equation*}
and we are looking for \(P=160\text{.}\) Replace \(P\) in the equation by 160 to get
Remember that we can always check our work by evaluating. Since \(T\) is measured in hours we need to go back and use \(T= 0.8\) hours, not 48 which is in minutes. Evaluating in our original function we get
Set up and solve an equation to find the number of bags of grass seed being carried by the truck if its total weight is 8 tons. Remember, “set up and solve an equation” means to use the Shoes and Socks Principle.
Is laughter really the best medicine? A study examined the impact of comedy on anxiety levels. Subjects’ anxiety levels were rated on a scale of 1 to 5 before and after the study. Levels averaged 4.3 before the study. There was no significant change in subjects in the control group. Subjects who watched the comedy videos showed a noticeable difference, and it depended on the hours of comedy watched. Anxiety levels fell an average of 0.098 (on the 1 to 5 scale) for each hour of comedy watched.
Make a table showing average anxiety levels for subjects who watched comedy videos for 0 hours (control group), 2 hours, 10 hours, and 20 hours, according to these findings.
If Lizbeth was charged $53.55 for the box of truffles she sent her mom, how many truffles were there? Set up and solve an equation to answer the question.
Between e-mail, automatic bill pay, and online banking, it seems like I hardly ever actually mail something.
Aside
But for those times, I need postage stamps. The corner store sells as many (or few) stamps as I want for 44¢ each but they charge a 75¢ convenience fee for the whole purchase.
Pecan pie? Yum. Not fitting into your favorite jeans? No fun. How far does Gretchen have to walk to burn off the calories from those two slices of pecan pie she ate last night? Each slice has approximately 456 calories.
The more expensive something is, the less likely we are to buy it. Well, if we have a choice. For example, when strawberries are in the peak of season, they cost about $2.50 per pint at my neighborhood farmer’s market and demand is approximately 180 pints. (That means, people want to buy about 180 pints at that price.) We approximate that the demand, \(D\) pints, depends on the price, $\(P\text{,}\) as described by the equation
Make a table of values showing the demand for strawberries priced at $2.00/pint, $2.25/pint, $2.50/pint, $2.75/pint, $3.00/pint, $3.25/pint, $3.50/pint.
It’s been a great week for strawberries and there are 240 pints to be sold at my neighborhood farmer’s. What price should the farmer charge for her strawberries in order to sell them all? Estimate your answer from the graph. Then set up and solve an equation to answer the question.
The stretch of interstate highway through downtown averages 1,450 cars per hour during the morning rush hour.
Aside
The economy is improving (finally), but with that the county manager predicts traffic levels will increase around 130 more cars per hour each week for the next couple of weeks. Earlier we found the equation
Significant slowdowns are expected if traffic levels exceed 2,000 cars per hour. When do they expect that will happen? Set up and solve an equation. Don’t forget to check your answer by evaluating.
If traffic levels exceed 2,500, the county plans to install control lights at the on-ramps. When is that expected to happen? Set up and solve an equation. Don’t forget to check your answer by evaluating.
