A caterer charges a fixed cost per party, plus an additional charge per guest. If it costs $300 to serve 25 guests and $420 to serve 40 guests, find the fixed cost and the cost per guest.
This is an example of a problem that can be solved using two variables and two equations.
The variables can be any variable. As we are going to solve by graphing, let’s use x and y; x for charge per guest and y for fixed cost.
Fixed cost
# guests
(charge per guest)
Total cost
Equation
y
25x
300
y + 25x = 300
y
40x
420
y + 40x = 420
Example 1: Solving by Graphing
Solve the system to find the fixed cost and charge per guest.
y + 25x = 300
y + 40x = 420
In slope-intercept form,
y = 300 – 25x
y = 420 – 40x
Graphing,
The solution point is (8, 100).
Thesolution to the problem is: the caterer charges $100 as a fixed price with an additional $8 per guest.
Quick Practice
At some point, most families have to decide whether or not to buy convenience items for the home. One of these items is a clothes dryer. One must consider cost, convenience, repairs, time factors, etc., when deciding.
Let’s say that the electric clothes dryer you are interested in buying costs $300 with tax.
It uses $0.50 of electricity for one hour of drying.
What is the rate of electric use (slope)?
0.50
What is the beginning cost (y-intercept)?
300
Use your answers to numbers 1 and 2 to write a linear equation in y = mx + b form.
Y = 0.50X + 300
It costs $1.50 per hour to dry clothes at a laundromat. The rate per hour is $1.50 (slope) and the beginning point is 0 (y-intercept). Write a linear equation in y = mx + b form to represent the total cost.
Y = 1.50X + 0 OR Y = 1.50X
Graph the two equations, using the graph below, and find the common solution point. What is the solution point?
When graphing, you may want to convert the slopes to more useable forms. For the first equation, m = 0.50 = 50/100, which would work well on the graph.
For the second equation, m = 1.50 = 150/100.
The x-value represents the number of loads it would take for the costs to be the same and the y-value represents the cost where they are the same.
Fill in:
It would take_____ loads at a cost of $_____ for the dryer at home to cost the same as the dryer at a laundromat.
Three hundred loads at a cost of $450
Fill in the table below. You may want to go back and read the beginning for ideas. Put down any point you can think of.
Advantages of Buying the Dryer
Advantages of Using a Laundromat Dryer
Advantages of Buying the Dryer
Advantages of Using a Laundromat Dryer
It is more private at home.
It is convenient to use at home.
You can dry more than one load at a time.
You don’t have to have change to use it.
You don’t have to pay for the electricity.
You can use any hour of the day and don’t have to leave home to use it.
It doesn’t cost a lot at the beginning.
You don’t have to share it with others or wait to use it.
You don’t have to pay for repairs.
Decision Time! Tell whether or not you have decided to buy the dryer. Give several reasons for your decision.
Any reasonable answer will do. Some possible answers:
You may decide based on what you think is more important. A possible answer might be: I decided that I didn’t want to put out the initial money for buying a dryer. It would be better for me to just spend the smaller amount needed to use the laundromat once a week. In addition, I wouldn’t have room to put the dryer in my home.
Alternately, you might say: I decided that it would be better to have a dryer in my home so that I didn’t have to go out to do the laundry. The initial cost is no problem, as I would make that back in the money I save each week not going to the laundromat. This way I can do the laundry and still be able to do other things at home while it is drying.
Activity 1( 20 points)
Answer each of the problems below to prepare you for 6-Activity 1. Then, click "Solving Systems" and complete 6-Activity 1.
The Bread Machine
In life, you must often make choices about whether to buy something pre-made or make it yourself. There are many things to consider: quality of homemade vs. bought, expense, convenience, enjoyment of making something, etc. In this activity, you will be looking at the choice of buying a bread machine or relying exclusively on store-bought bread.
The bread machine you are interested in costs $100 with tax. The ingredients to make one loaf of bread cost $0.80. What is the rate of cost of one loaf of bread?
What is your start up cost (cost of machine)?
Write a linear equation, y = mx + b for the total cost.
A loaf of bread of similar quality in the store costs $1.60. This is the rate for a loaf of bread (the slope). As there is no other cost, your beginning point (y-intercept) is zero. Write the linear equation y = mx + b for the total cost.
Graph the equations on a graph like this. You may use either point plotting or slope-intercept. Be sure to locate at least three points. You may want to do this in pencil in case you decide to use more points later in the problem.
Estimate the ordered pair, where the two lines cross, and write it here: (______, ______)
The x-value, where they cross represents the number of loaves it will take to make the costs of the bread machine and buying the bread equal. The y-value represents the cost where they are the same. Fill in:
It would take _________ loaves at a cost of $_________ for the bread-maker and store bought bread to cost the same.
Fill in the table below. You may want to go back and read the beginning for ideas. (And yes, if you like the way it smells when it bakes, that’s an advantage.) Write down any point you can think of.
Advantages of Buying the Machine
Advantages of Buying at the Store
Decision Time! Tell whether or not you have decided to buy the bread machine. Give several reasons for your decision.
Section Homework 1 (10 points)
It’s time to do your homework. You may take this only one time, so check your understanding of the material before doing your homework, and do your best.
Activity 1(
Section Homework 1 (