week 14
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Massachusetts Bay Community College *
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Mathematics
Date
Feb 20, 2024
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docx
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Turn It In Systems of Linear Equations
Name:
Answer the following:
We now consider how two or more linear functions interact with each other. As consumers, we are often faced with choices. In this lesson, we will solve systems of equations in order to determine the best deal for our needs.
Much of mathematics work we have done this semester has depended on making connections among representations. To study and solve systems of equations, the interrelationships between tables, graphs, and equations are critical. Think about the contexts, and create tables, graphs and equations to extend what you have learned about linear functions to work with them two or more at a time.
Fitness Plans
By now, you are proficient at finding equations. You will use tables to locate an interval
over which two or more of the fitness plans cost the same amount. You will refine your search for an intersection point with graphs and then by solving pairs of equations.
1. Consider payment plans for three different fitness centers:
Shape Up
charges an initiation fee of $39 plus $35 per month per person.
Fitness Fun
charges an initiation fee of $75 plus $20 per month per person.
Be Your Best
has no initiation fee and charges $52 per month per person.
Complete the table showing the total cost for the first 6 months for each fitness
company's payment plan (initiation fees are only charged once.)
Amount Paid so far (in dollars)
Month
Shape Up
Fitness Fun
Be Your Best
1
74
95
52
2
109
115
104
3
144
135
156
4
179
155
208
5
214
175
260
6
249
195
312
x
2. For each set of data, make a graph (month, amount paid so far). Label each graph
with the company's name.
Shape Up
Fitness Fun
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Be Your Best
3. a. Which company charges the most after 6 months of membership? How do you
know?
312 the largest of three values 249, 175, 312. Be Your Best charges most.
b. How can you tell from a graph?
For the third graph the value ordinate for the abscise x=6 is the largest
c. How can you tell from a table?
In the raw correspondent to 6 months the last value is the largest
d. Which company has the steepest graph?
Be Your Best Why does that make sense?
The steepest graph is for the function which grows most quickly. Since it grows quickly,
for the same values of x the values of y are larger starting from some value.
4. What does the y-intercept represent for each fitness payment plan?
It represents an initial fee
5. Find an equation that fits each data set. Write your equation in the form y = mx + b.
Include the equation in the last row of the table.
Shape Up
Fitness Fun
Be Your Best
6. You have $500 to spend on a fitness plan this year. Show your work.
a. Which fitness plans can you afford and why?
Shape Up
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Fitness Fun
Be Your Best
b. For how many months can you afford each fitness plan?
Shape Up
Fitness Fun
Be Your Best
7. a. Solve each equation you found in problem 5 for the variable that represents
months.
Which variable in your equations represents months?
x
Shape Up
Fitness Fun
Be Your Best
b. What do these equations allow you to do directly that the equations in problem 6
require more work to do?
Calculate the number of months for the given cost.
8. Consider the following pairs of fitness plans. For what number of months will you
pay the same amount for a membership in either plan? Be sure you’ve read or watched
the material on solving systems of equations. You want to make a system of equations
out of the equations for each pair of fitness plans. Show your work.
Shape Up
and
Fitness Fun:
Fitness Fun
and
Be Your Best:
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Be Your Best
and
Shape Up:
BEYOND FITNESS PLANS
9.
Study the lines.
a.
Estimate the intersection point of the lines.
Approximately b.
Find at least two grid points on each line. Do not estimate.
c.
Find an equation for each line. Equation 1
Equation 2
c.
These two equations together is called a system of equations in two unknowns
because the values of x and y of the intersection point are both unknown. Solve
the system algebraically. Show your work. Find both x and y and write your
answer as an ordered pair.
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e.
Compare your answers in problems 9a and 9d. Is your estimate reasonable?
10.
Graph the following equations.
y = 2x - 3
y = -0.5x + 4
a.
Estimate the intersection point graphically. Include the graph below.
b.
Solve the problem algebraically. Show your work. Write your answer as an ordered
pair.
11.
In the cartoon, Foxtrot, Paige is frustrated by a system of equations problem she has
to solve for homework
(See https://www.gocomics.com/foxtrot/2009/01/25/
, retrieved April 23,
2020.)
The systems is:
2x + y = 60
x + 2y = 75
Her older brother asked her, "If 2 shirts and a sweater costs $60 and a shirt and 2
sweaters costs $75, what does each item cost?" Paige solved the problem immediately
but didn't realize she had done so.
a.
What was Paige's answer? (You can find it in the cartoon).
A shirt costs 15 and a sweater costs 30
b.
Show Paige how to solve the system of equations in her homework. Use two
different methods to solve the problem. This video
or this video
might help.
Method 1
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Method 2
12. The admission fee at the Grand Rapids Public Museum is $3 for children and $8 for
adults. One day 575 people paid to enter the museum. The museum collected $3450 for
admissions.
a. Fill in the first two rows of the table.
Quantity
Rate
Value
Children
3
Adults
8
Total
Initially, the number of Children and the number of Adults are unknown.
Choose
variables
to
indicate
each
of
these.
The rates (ticket prices) are given in the problem.
How can you use each Quantity and Rate to get the Value? Multiply Quantity by
Rate to get the Value.
b. Fill in the empty cells in the last row of the table. The Total Quantity and the Total
Value are both given in the problem.
c. Write an equation showing the relationship between the Quantity of Children, the
Quantity of Adults, and the Total Value.
d.
Write an equation showing the relationship between the Value of Children's
tickets, the value of Adults' tickets, and the total value.
The equations you wrote in problems 2c and 2d provide a system of linear equations.
e. Solve the system in two different ways. Verify that your solution works.
Method 1
Method 2
f. How many children and how many adults paid for admission to the museum that
day?
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230 children, 345 adults
13. Melanie's favorite ways to exercise are playing basketball and running. Her goal is
to exercise 60 minutes each day, splitting her time between both basketball and
running. She also wants to achieve the equivalent of 10,000 steps per day. Playing
basketball for one minute is equivalent to taking 130 steps/minute. Running at the rate
of 5 miles per hour is equivalent to taking 185 steps per minute.
a. Use the table to set up two equations, one showing the relationship between the
number of minutes Melanie will play each sport, and one showing the relationship
between the equivalent numbers of steps Melanie will take during each activity.
Quantity
Rate
Value
Playing Basketball
130
Running
185
Total
b. Solve the system of equations using two different methods. Method 1
Method 2
c. How many minutes of basketball, B, and minutes of running, R, should Melanie do
each day to exercise 60 minutes and take the equivalent of 10,000 steps?
20 minutes of basketball and 40 minutes of running
14. Filiz's favorite ways to exercise in the summer are walking and cycling. Her goal is
to exercise 90 minutes each day, splitting her time between both types of exercise. She
also wants to achieve the equivalent of 12,000 steps per day. Walking for one minute at
the rate of 3.5 miles per hour is equivalent to taking 130 steps/minute. Cycling at the
rate of 15 miles per hour is equivalent to taking 160 steps per minute.
a. Use the table to set up two equations, one showing the relationship between the
number of minutes Filiz will engage in each type of exercise and one showing the
relationship between the equivalent numbers of steps Filiz will take during each
activity.
Quantity
Rate
Value
Walking
130
Cycling
160
Total
b. Solve the system of equations using two different methods.
Method 1
Method 2
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b.
How many minutes of walking, W, and how many minutes of cycling, C,
should Filiz do each day to exercise 90 minutes and take the equivalent of 12,000
steps?
80 minutes of walking and 10 minutes of cycling
c.
What did you learn about using Quantity Rate Value Tables? How can you use
what you learned to set-up systems of equations problems?
We should write values in the table and then set up a system by the values given
in the condition.
15.
Use what you learned about Quantity Rate Value Tables (QVR Tables) to solve this
problem.
Gummy candy costs $5 per pound. Taffy candy costs $4 per pound. You want to buy 4
pounds of candy and spend exactly $17.50.
Quantity
Rate
Value
Gummy candy
5
Taffy candy
4
Total
a.
Fill in the first two rows of the table using the information from the story.
b.
Define your variables. Be specific about what each one means.
d.
Write two equations that arise directly from the table.
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d.
Solve the system of equations. Use two different methods. Show your work. Verify
that your solution works.
Method 1
Method 2
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e.
Interpret your solution based on the problem with gummy and taffy candies.
You should buy 1.5 pounds of Gummy Candy and 2.5 pounds of Tuffy Candy to have 4
pounds of candy which cost exactly $17.5
Watch
this video
or
this video
or
this video
to see if it helps.
Write your answers to the above questions in one file (recommended .doc/.docx or
PDF; not pages).
Make your work neat and easy to follow.
Answer any questions that
require written work with complete sentences.
Include pictures or illustrations where
requested.
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