Section6.4Interlude: Point-slope equation of a line
SubsectionLearning to make pasta
Yarra and Esti want to do something fun (and a bit extravagant) to celebrate their tenth wedding anniversary. They want to invite friends to come with them to a local cooking school to learn how to make pasta.
The website says a 2-hour class including instruction and a full meal for 6 people costs $739 and for 10 people it costs $1000. Yarra and Esti decide to go for it and invite friends so they’ll have a group of 10.
But, that gets Yarra wondering what the equation was for cooking class pricing. She assumes it’s a linear equation with some fixed costs and then a per person rate.
\begin{align*}
C \amp = \text{ cost of cooking class (\$) }\sim \text{ dep}\\
P \amp = \text{ total number of people (people) }\sim \text{ indep}
\end{align*}
Yarra knows the next step to write the equation is to find the intercept (fixed cost) but Esti said there’s another way to write the equation that’s even easier.
Check this out. Esti shows her how to figure out what 15 people will cost without finding the intercept as an example. First, she calculates that it’s \(15-10=5\) people more than 10 people. She reminds Yarra that the slope is $65.25/person, so the cost will be \(5 \times 65.25 = \$326.25\) more than the 10-person cost. Since the 10-person cost is $1000, Esti figures out that the 15-person cost should be \(\$1000 + \$326.25 = \$1326.25\text{.}\)
Let’s take a closer look at what Esti did to get the cost for 15 people because that process will work in general. Her entire calculation in one line was
Yarra has several questions about Esti’s form of the equation. Her first question is if there’s any easy way to find the intercept (upfront cost) from Esti’s form to the standard equation of a line. Yes, there is. The fixed cost is the value of \(C\) when \(P=0\) which is
Notice that this is not the same as the cost of having 0 people which would be $0, but maybe it’s the cost of reserving for a party and then cancelling?
Yarra’s other question is what if Esti had used the other known pair instead. For example, if she used that 6 days cost $739, then the equation would be
If you see these equations in a precalculus or calculus course, those textbooks (and professors) often use \(x\) for the independent variable and \(y\) for the dependent variable. The constants are \(m\) for the slope and \(b\) for the intercept. The point is often called \((x_{1},y_{1})\) which is a little confusing because they look like variables, but the subscripts indicate that we mean a specific value of \(x\) and \(y\text{.}\) In that notation, these equations can be written
Maryn is very happy. Her interior design business is finally showing a profit. She has logged a total of 471 billable hours at $35 per hour since she started her business. Accounting for start-up costs, her profit is $2,194.
Huiyi is conducting an experiment for his biopsychology class. He measures each participant’s accuracy in completing a complicated task involving working memory when visual distractions (flashing lights) are introduced. Participants with 5 flashes of light scored an average of 82 points on the task. Huiyi has read that each additional flash of light should decrease the average by 3.5 points.
If what Huiyi read is correct, what would be a linear equation he can use to predict average score as a function of flashes of light? Use the point-slope equation and don’t forget to name the variables, including units.
Thanks to conservation laws that protect bamboo forests and crackdowns to stop poaching (illegal killing), Giant Pandas are no longer considered an “endangered” species (although they are still “vulnerable to extinction” ). The Giant Panda population in the wild is growing exponentially, but the growth is slow enough that we can approximate the growth with a linear equation. There were approximately 1596 Giant Pandas in 2004 and around 1864 Giant Pandas in 2025.
Sound Transmission Class (STC) of an interior wall (like in your apartment) corresponds roughly to how many decibels of noise the wall blocks. Higher numbers mean more sound is blocked. A basic interior partition wall with a single layer of drywall and standard insulation with a density of 11 kg/m³ provides an STC of 39. By upgrading to a dense 35 kg/m³ mineral wool core, the STC jumps to 50, drastically lowering the transmitted noise level.
Haajira is knitting a scarf. She’s worked on it for two hours and has knit 10″ so far. It took her some time to get everything set up, but now that she’s going, she expects to knit about 7″ per hour.
Nyong is going to visit friends for a week. He has a flight but has not yet made arrangements to get a rental car. He’d like to rent a midsize SUV because he and his friends are going on a golf trip for the first 3 days. He’s not sure if he should just rent the car for the full week. Nyong looked online and found 3 days will cost $295 and the full week cost $444, both prices include all taxes and fees. Assume that cost increasing linearly in the number of days.
Ecologists studying mountain ecosystems observe that trees grow shorter at higher elevations. A research team found that at an elevation of 1,200 feet, the canopy height is around 150 feet. Their data show that above this elevation, the canopy height decreases at a constant rate of 0.02 feet for every additional foot of elevation.
Using the distributive property and algebra, rewrite the equation in slope-intercept (standard) form. Hint: make sure your equation includes your answer to (c).
A citizen’s group is working to register people to vote across the county. After 3 weeks, they were able to lower the unregistered voters rate to 36%. Two weeks later (after 5 weeks total), they were able to lower the unregistered voters rate to 28%.
Zaneta and Melia are so excited! It’s time for the Minnesota State Fair. They are especially looking forward to riding the Midway rides. They can buy a Fun Card for $44 which is worth 40 tickets or Fun Card for $70 which is worth 80 tickets. (So you know, each ride itself costs 4-8 tickets so they use up tickets pretty quickly.) Assume that the price of a Fun Card is a linear function of the number of tickets it’s worth.
