Algebra 1, Phone Smarts Name_______________________________
Per _______ Date____________
In this investigation, you will work on fast methods for finding solutions to systems of equations.
Recall that Jack Bauer in “Phone Plan” had a choice between AT&T and Verizon. The following system of equations describe the monthly costs.
 C
  m
 C 14 0.10
m where C is monthly cost ($) and m is the number of minutes of talk time. We call this a system of equations because multiple equations are considered at the same time.
R EVIEW
1.
Which of the two equations in the system represent AT&T’s monthly cost? (Hint: AT&T charges 10 cents per minute of talk time.)
2.
What does the equation 8 + 0.15m = 14 + 0.10m mean in the context of the situation?
3.
Do you remember that both companies charge the same monthly amount ($26) if Jack talks 120 minutes?
Describe how you could use a table or graph to find this solution to the system.
T HINK
4.
Although tables and graphs are great for solving problems, many people find algebra to be faster.
Try your hand at solving 8 + 0.15m = 14 + 0.10m using algebra. (Some hints are given below.)
Original equation 8 + 0.15m = 14 + 0.10m
Subtract 8 from both sides
Subtract 0.10m from both sides
5.
You should have found m = 120 is the solution in your algebra work above.
What does m = 120 mean for this situation?
6.
What do you get when m = 120 is put into the Verizon cost rule? The AT&T rule?
7.
In your own words, state the solution to the system.

To summarize our work so far, we call the solution to the system the values for variables that satisfy both equations.
Depending on your approach, the solution will be seen differently.
8.
Identify (circle or whatever) the solution to the system in each of the method below.
9.
Consider the system  y
 y
2
3 x x
. a.
Graphically. Create a graph to verify that x= 3 and y = 5 is the
solution to the system. b.
Numerically. Create a table to show that x= 3 and y = 5 is the
solution to the system. c.
Analytically. Use substitution to verify that x= 3 and y = 5 is the solution to the system. That is, plug in values for the variables and show they work in both equations.

Let’s look closely at the method of solving systems of equations using algebra.
10.
Suppose Jack Bauer and the Counter Terrorist Unit need to purchase a new satellite (GPS) pinpoint service.
One company, Big Brother, charges $15 a month after a $200 startup fee.
Another company, Eye in the Sky, charges $25 a month after a $100 startup fee.
The system
 C
 C

15 m

200
25 m

100
represents this situation where C is total cost ($) and m is the number of months. a.
Write the equation that states the costs for each company are the same. b.
Use algebra to solve for m in your equation in part a. c.
If Jack and the Counter Terrorist Unit wanted to use Big Brother, what would be the cost for the number of months you found in part b? d.
What would the cost be if they used Eye in the Sky for that number of months? e.
Are your answers from c and d the same? Why do you think they should or shouldn’t be the same? f.
State the solution to the system and explain what it means in the context of this situation.
11.
Consider the system
 y
 y

60 4 x x
which might describe some cost by two different companies.
Where do you start when solving systems algebraically? Take a look at two different thoughts.
Discuss both types of thinking with a classmate. Which one makes more sense to you?

12.
Algebraically find the solution to the system

 y y

60 4 x
. (Hint: Look at #11 to help you get started.)
80 6 x
13.
Verify your solution in the last problem analytically. (Look back at #9 if you need.)
14.
Practice solving the following systems with algebra.

 y y
  x
2( x 12)

 y y 2 x x
3
 

 z
1
2
(4
8(6 x

 x )
6)
15.
Verify your solutions in the last problem analytically.
 b b
2
3 a
(

2 a
3
5) 2

1

16.
To dispose of nuclear material, Jack Bauer and the Counter Terrorist Unit received the following quotes from two companies:
 C
  m
where m is the amount of nuclear
 C
 m material in grams and C is the cost in thousands of dollars. a.
Without solving, do you think that the costs for the two companies will ever be the same? Explain your reasoning. b.
Without graphing, what do you think will be true about the two lines on a graph? Explain your thinking. c.
Try to solve the system
 C
 C
  m
 m
using algebra. Something very strange will happen. Be prepared to discuss your results with the class. d.
Are there values of m for which the costs be the same? e.
Would you consider this system of equations to have “one solution”, “infinite solutions”, or “no solution”?
17.
Suppose two companies gave you the following costs functions to do work on some project:



 x
 
  x

6( x

1)
. a.
Without simplifying, look at both cost functions in a table on your calculator. What do you notice? b.
c.
For what values of x will the costs be the same for the two companies?
Would you consider this system of equations to have “one solution”, “infinite solutions”, or “no solution”? d.
Rewrite both cost functions by simplifying the expressions to show why this system has the type of solution that it does.
18.
For each of the following, create your own example of a system of equations that has
One solution No solution Infinite solutions

19.
Suppose Jack Bauer wanted to buy treats for his team. He has $140 to spend on scones ($3 each) and muffins ($2 each). Jack also knows his team usually eats about twice as many muffins as scones. The following system models this situation.


3 m s

2
2 s m

140
where m is the number of muffins and s is the number of scones Jack buys. a.
If Jack buys 6 scones and 12 muffins, which of the equations is satisfied? b.
If Jack buys 10 scones and 55 muffins, which of the equations is satisfied? c.
To solve this system of equations consider the thinking below.
We call this method “substitution” since the m is replaced with what it equals. To “substitute” means to take the place of another, so in the substitution method we replace a variable in one equation with what we know from the other equation.
Try to use the substitution method to find the solution to the system


3 m s

2
2 s m

140
. d.
Verify your solution in the last part. e.
Explain what the solution to the system


3 m s

2
2 s m

140
means in the context of this situation.
20.
Practice solving these systems.


5 x

2 y
 
26 x 3 y
  w

2.5
h
 h w
 4 x

5
2 y x
 
1
51