Name____________________________ Period______ Date________
Linear Functions – Activity 3.3 – Systems of Equations
Moving Out: Finals are over and you are moving back home for the summer. You need to rent a truck to
move your possessions from the college residence hall back to your home. You contact two local rental
companies and obtain the following information for the 1 day cost of renting a truck:
Company 1: $19.99 per day plus $0.79 per mile
Company 2: $29.99 per day plus $0.59 per mile
The total cost of renting a truck for 1 day is a function of the number of miles driven.
1. Identify the input and output in this situation.
Input___________________
Output__________________
2. Let 𝑛 represent the total number of miles driven in 1 day.
a. Write an equation to determine the total cost 𝐶 of renting a truck for 1 day from company
1.
b. Write an equation to determine the total cost 𝐶 of renting a truck for 1 day from company
2.
3. a. Complete the following table to compare the total cost of renting the vehicle for the day. Verify
your results using the Table feature of your graphing calculator.
𝑛, Number of miles driven
Total cost, 𝐶, Company 1 ($)
Formula
0
10
20
30
40
50
60
70
80
Total cost, 𝐶, Company 2 ($)
1
b. For what mileage is the 1-day rental cost the same?
c. Which company should you choose if you intend to drive less than 50 miles?
d. Which company should you choose if you intend to drive more than 50 miles?
4. a. Graph the two cost functions, for 𝑛 between 0 and 120 miles, on the same coordinate axes. Make
sure you label the axes and title the graph. Use 2 colors or dashes and a solid line to identify the 2
graphs.
b. Use the table from Problem 3a to determine the point where the lines in part 𝑎 intersect.
What is the significance of the point in this situation?
Definition - System of Linear Equations
A linear system of 2 equations in 2 variables can be written:
𝑦 = 𝑎𝑥 + 𝑏
𝑦 = 𝑐𝑥 + 𝑑
A solution of the system, if it exists, is the ordered pair (𝑥, 𝑦) that makes both equations true. This is
Sometimes,
it is advantageous
tohas
rewrite
a linear
equationthe
given
the is
the point on a graph where the
2 lines intersect.
If the system
exactly
one solution,
system
calledbstandard
consistent.form as its equivalent slope-intercept form. For example, consider the equation 3𝑥 +
7𝑦 = 5. To write this equation in slope-intercept form, you need to solve for 𝑦 as follows.
2
The system in the truck rental situation can be written
𝐶 = 19.99 + 0.79𝑎
𝐶 = 29.99 + 0.59𝑎
You first solved the system numerically in Problem 3 by completing the table and noting the value of the
input that produced the same output. You then solved the system graphically in Problem 4 by locating
the point of intersection of the two lines.
Substitution Method
You can also use algebraic methods to solve systems of equations. One algebraic model is the
substitution method. To use the substitution method, you solve one of the equations for one of the
variables and then substitute for that variable in the other equation. The following example illustrates
this method:
Solve the following system algebraically using the substitution method.
𝑦 = 3𝑥 − 10
(1)
𝑦 = 5𝑥 + 14
(2)
SOLUTION
Since the equation (1) is solved for 𝑦, substitute 3𝑥 − 10 for 𝑦 in equation (2) to get
3𝑥 − 10 = 5𝑥 + 14
To solve the equation for 𝑥, you need to isolate the variable. Rewrite the equation so that all terms
involving 𝑥 are on one side and all other terms are on the other side. This is generally accomplished by
adding and/or subtracting the appropriate terms to/from each side of the equation.
To determine the corresponding 𝑦-value, substitute −12 for 𝑥 into either of the original equations and
solve for 𝑦. Substituting into equation (1),
𝑦 = 3(−12) − 10 = −36 − 10 = −46
3
Remember that the ordered pair (−12, −46) represents the point of intersection of the two lines as
well as the solution to the system This is verified by the following screen:
For the truck rental problem, you want 𝐶 (total cost) to be the same for both functions. Because each
expression is solved for 𝐶, set the expressions equal to each other. You are substituting an expression
for 𝐶 from one equation into the other equation.
5.
a. Solve the introductory problem involving the 1-day cost of renting a truck by solving the
following system for 𝑛 using the substitution method.
𝐶 = 19.99 + 0.79𝑎
𝐶 = 29.99 + 0.59𝑎
b. What does the value of 𝑎 represent in part a?
c. Determine the value of C by substituting the value of 𝑎 from part a into one of the original
equations and solving for C.
d. Check to see if your solution satisfies both equations. Substitute into the other equation
4
6. You are going to graduate and are interested in purchasing a new car. You have narrowed the choice
to a Honda Accord LX and a Passat GLS. You are concerned about the value of the car depreciating
over time. You search the Internet and obtain the following information. The depreciation per year
is the amount by which the value of the car will decrease each year.
a. Complete the following table.
0
1
2
3
4
b. Will the value of the Passat GLS ever be lower that the value of the Accord LX? Explain.
c. Let 𝑉 represent the value of the car after 𝑥 years of ownership. Write an equation to
determine 𝑉 in terms of 𝑥 for the Accord LX. Write another equation to determine 𝑉 in
terms of 𝑥 for the Passat GLS.
d. The two equations in part c form a 2x2 system of linear equations. Solve the system using
the substitution method.
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e. The solution to the system is an ordered pair of the form (𝑥, 𝑉). What do the values 𝑥 and 𝑉
represent in this solution?
f.
What car has the better resale value? How can you tell from the graph? Explain
Types of Linear Systems
7. You and your friend are traveling in the same direction, but in different cars, on the New York State
Thruway.
a. Let 𝑑 represent the distance you are from the common starting point. Write an equation for
𝑑 in terms of time, 𝑡, if you are traveling at 65 miles per hour.
b. Write another equation to determine the distance 𝑑 in terms of time, 𝑡, that your friend is
from the starting point. She is traveling at 60 miles per hour and has a 30 mile head start.
8. The linear equations in Problem 7 form a 2x2 system of linear equations:
𝑑 = 65𝑡
𝑑 = 30 + 60𝑡
a. Solve the given system using each of the following methods.
Numerically:
t
𝑑 = 65𝑡
𝑑 = 30 + 60𝑡
(IN HOURS)
YOUR DISTANCE (miles)
YOUR FRIEND’S DISTANCE (miles)
0
1
2
3
4
5
6
6
Graphically
Algebraically – solve for both variables
b. What is the solution of the system? What is the meaning of this solution in the context of
this problem?
Definition - Consistent System of Linear Equations
A linear system with exactly one solution.
Do all systems have exactly one solution? Consider the following.
9. a. You and your friend are traveling again. She still has a 30-mile head start, but this time both of
you are traveling at 60 miles per hour. When will you catch up with your friend? Sketch a picture
and explain.
b. The situation just described can be represented by the following system and the
corresponding graph:
𝑑 = 60𝑡 + 30
𝑑 = 60𝑡
7
Do the lines intersect? What is the solution to the system?
b. Try to solve the system algebraically using the substitution method. What type of equation do
you obtain?
The linear system in problem 9 is said to be inconsistent. There is no solution because the lines
never intersect. Graphically the slopes of the lines are equal, but the vertical intercepts are
different. Therefore the graphs are parallel lines. Solving such an equation algebraically results in
a false equation such as 30 = 0.
8
10. a. You and your friend are taking one more trip. This time she does not have a head start. You both
leave from your house, both travel in the same direction, and both travel at 60 miles per hour.
When will you both be at the same point?
b. The system for this situation is
𝑑 = 60𝑡
𝑑 = 60𝑡
Try to solve this system graphically. Do the lines intersect? What is the solution to the system?
c. Try to solve this system algebraically. What type of equation do you obtain?
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The linear system in problem 10 is an example of a dependent system. Graphically, in such a
system, both equations represent the same line. The system has an infinite number of solutions.
Solving a dependent system algebraically results in an equation that is true for all t, such as 0 =
0 or 1 = 1.
Summary - System of Equations
1. A 2 × 2 system of linear equations consists of two equations with two variables. The graph of each
equation is a line.
2. A solution to a 2 × 2 system of equations is an ordered pair that satisfies both equations of the
systems.
3. Solutions can be found in three different ways:
a. Numerically, by examining tables of values for both functions.
b. Graphically, by graphing each equation and finding the point of intersection.
c. Algebraically, by combining the two equations to form a single equation in one variable, which
can then be solved. This is called the substitution method.
4. A linear system is consistent if there is at least one solution, the points of intersection of the
graphs.
5. A linear system is inconsistent if there is no solution; the lines are parallel.
6. A linear system is dependent if there are infinitely many solutions. The equations represent the
same line.
7.
8.
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Practice
1. Solve the systems of linear equations numerically, graphically, and algebraically (substitution
method). The numerical table will help you graph the systems.
a. 𝑦 = 2𝑥 + 3 𝑦 = −𝑥 + 6
Numerically
𝑥
−2
−1
0
1
2
3
Graphically
𝑦1
Algebraically
𝑦2
b. 𝑦 = 0.5𝑥 + 9 𝑦 = 4.5𝑥 + 17
Numerically
𝑥
−2
−1
0
1
2
3
Graphically
𝑦1
Algebraically
𝑦2
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c. 𝑦 = 5𝑥 − 3 𝑦 = 5𝑥 + 7
Numerically
𝑥
0
1
2
3
4
Graphically
𝑦1
Algebraically
𝑦2
2. Two companies sell software products. In 2010, company A had total sales of $17.2 million. Its
marketing department projects that sales will increase by $1.5 million per year for the next several
years. Company B had total sales of $9.6 million of software products in 2010 and projects that its
sales will increase by $2.3 million each year.
Let 𝑛 represent the number of years since 2010.
a. Write an equation that represents the total sales, 𝑠, of company A since 2010.
b. Write an equation that represents the total sales, 𝑠, of company B since 2010.
c. The two equations in parts a and b form a system. Solve this system to determine the year
in which the total sales of both companies will be the same.
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3. You are considering installing a security system in your new house. You gather the following
information from two local home security dealers for similar security systems:
Dealer 1: $3560 to install and $15 per month monitoring fee.
Dealer 2: $2850 to install and $28 per month monitoring fee.
Although the initial fee of dealer 1 is much higher than that of dealer 2, dealer 1’s monitoring fee is
lower.
Let 𝑛 represent the number of months you have the security system.
a. Write an equation that represents the total cost, 𝑐, of of the system with dealer 1.
b. Write an equation that represents the total cost, 𝑐, of of the system with dealer 2.
c. Solve the system of equations that results from parts a and b to determine the number of
months for which the total cost of the systems will be equal.
d. If you plan to live in the house and use the system for 10 years, which system would be less
expensive? (Hint – convert 10 years to months).
4. You can run a 400-meter race at an average rate of 6 meters per second. Your friend can run the
race at a rate of 5 meters per second. You give your friend a 40-meter head start. She then runs 360
meters.
a. Write an equation for your distance, in meters, from the starting point as a function of time,
in seconds.
b. Write an equation for your friend’s distance, in meters, from the starting point as a function
of time, in seconds.
c. How long does it take you to catch up to your friend?
d. How far from the finish line did you meet? (you will calculate how far you ran so figure out
how to answer this).
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5. For many years, the life expectancy for women has been longer than the life expectancy for men. In
the past few years, the life expectancy for men has been increasing at a faster rate than that for
women. Using the data from the Centers for Disease Control and Prevention and the National
Center for Health Statistics, the life expectancy E, for men and women in the United States can be
modeled by:
Women: 𝐸 = 0.115𝑥 + 77.42
Men: 𝐸 = 0.212𝑥 + 69.80
Where 𝑡 represents the number of years since 1980.
a. Solve the system numerically by completing the following table. Approximate the value of 𝑡
to the nearest year. Use your calculator.
t, NUMBER OF
LIFE EXPECTANCY
LIFE EXPECTANCY FOR MEN
YEARS SINCE 1980
FOR WOMEN
0
25
b. What does the solution to this system represent in the context of this problem?
c. The problem is displayed graphically.
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d. Solve the system algebraically using the substitution method.
6. You are the manager of a small company producing interlocking paving pieces, called pavers, for
driveways. You sell the pavers in bundles that cost $200, each bundle contains 144 pavers. The total
cost in dollars, 𝐶(𝑥), of producing 𝑥 bundles of pavers is modeled by 𝐶(𝑥) = 160𝑥 + 1000.
a. The revenue is the amount of money collected from the sale of the product. Write an
equation for the revenue function in dollars, 𝑅, from the sale of 𝑥 bundles of pavers.
b. Determine the slope and the 𝐶-intercept of the given cost function. Explain the practical
meaning of each in this situation.
c. Determine the slope and the 𝑅-intercept of the revenue function. Explain the practical
meaning of each in this situation.
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d. Graph the cost and revenue functions on the same set of axes. A company will break even
when its revenue, R, exactly equals its cost, C. Estimate the break-even point from the
graph. Express your answer as an ordered pair, giving units. Check your estimate of the
break-even point by graphing the two functions on your graphing calculator.
𝑥
0
10
20
30
40
50
𝑦=
e. Determine the exact break-even point algebraically. If your algebraic solution does not
approximate your answer from the graph in part d, explain why. Make sure you solve for
both variables.
f.
How many bundles of pavers does the company have to sell for it to break even?
g. What is the total cost to the company when you break even? Verify that the cost and
revenue values are equal at the break-even point.
h. For what values of x will your revenue exceed your cost?
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i.
As a manager, what factors do you have to consider when deciding how many pavers to
make?
j.
If you knew you could sell only 30 bundles of paper, would you make them? Consider how
much it would cost you and how much you would make. What if you could sell only 20?
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