Restaurant Receipts Cards - Highland Park ISD

Restaurant Receipts Cards 

Algebra 2 

HS Mathematics 

Unit: 05 Lesson: 01 

The following four cards serve as a group that can be used collectively to answer the question. 

Alice’s Diner sells only four items, which 

have individual prices of $2, $3, $4, or $5. 



Qty 

2 

3 

CUSTOMER RECEIPT 

Item 

Super Sodas 

Big Burgers 

Qty 

5 

6 


Item 

Super Sodas 

Big Burgers 

TOTAL: 

$18 

TOTAL: 

$39 

THANK YOU!! 

THANK YOU!! 

After dining there, Group A got this receipt. 

What is the cost of each individual item? 

After dining there, Group C got this receipt. 








Qty 

2 

Item 

Big Burgers 

Qty 

2 

Item 

Monster Nachos 

2 

Fast Fries 

3 

Big Burgers 

2 

Super Sodas 

2 

Fast Fries 

4 

Super Sodas 

TOTAL: 

$18 

TOTAL: 

$38 

THANK YOU!! 

THANK YOU!! 

After dining there, Group B got this receipt. 


After dining there, Group D got this receipt. 


©2010, TESCCC 08/01/10 page 9 of 60

Restaurant Receipts 

Algebra 2 



Fill in or complete the information on this page as you work through the problem. 

 

What information is found on each receipt? 

Receipt A 

Receipt C 

Receipt B 

Receipt D 

 

What prices are possible for each item? 

List possible item prices 

Check—Do the prices work? 

 

What prices can be eliminated for certain items? 

Write the names of the items 

in the first column. 

Write the possible costs for each item in the top row. 

Eliminate 

possibilities by 

placing X’s in the 

appropriate boxes. 

©2010, TESCCC 08/01/10 page 11 of 60

Systems of Equations 

Algebra 2 



Using B for the cost of a burger, F for the cost of fries, N for the cost of nachos, and S for the cost of a 

soda, write equations for each statement below. 

2 sodas and 3 burgers cost $18. 

5 sodas and 6 burgers cost $39. 

2 burgers, 2 fries, and 2 sodas cost $18. 

2 nachos, 3 burgers, 2 fries, and 4 sodas cost $38. 

DEFINITION: 

DEFINITION: 

When used together, these form a ________________________ of equations, 

which is a set of two or more equations with two or more unknowns (or variables). 

The ________________________ to a system lists the values that can be used 

for each variable to make the equations in the system true (or, “check”). 

Samples: 

1) 

2x 

3y 

4z 

3 

x z 4 

x 2y 

3z 

4 

This is a _______ _____________ of equations because it uses 

_____ _________________ and has _____ ________________. 

Which of the following would be the solution to this system? 

A) x 4, 

y 1, 

z 2 B) x 9.5, 

y 0, z 5. 5 

C) x 5, 

y 3, 

z 1 

2) 

2x 

7y 

y 

1 x 2 

9 

3 

This is a _______ _____________ of equations because it uses 

_____ _________________ and has _____ ________________. 

Which ordered pair shows the solution to this system? 

A) (15, 3) B) (8, 1) C) (2, -2) 

Below, circle the correct solution to each equation. How can you determine the solution? 

3) 

4) 5x 

3y 

2z 

16 

y 3x 

8 

x y 5 z 

x 4y 

1 

8x 

5y 

3z 

25 

A) x 3, 

y 1 

A) (1, -0.2, 5.8) 

B) x 7, 

y 29 

B) (-2, 4, 7) 

C) x 7, 

y 9 

C) (1, 1, 4) 

These answers are 

called “ordered triples” 

©2010, TESCCC 08/01/10 page 13 of 60

Plug into the System 

Algebra 2 



Evaluate each equation for the given values in the ordered triple. If the values “check” in the equation, 

then shade in the corresponding box. When all items are complete, the remaining boxes (which have 

not been shaded) will spell out a quote. Be sure to check each solution set for each equation. The 

equation will have more than one set that works. 

1) System of Equations: (-2, 3, -3) (2, 6, -4) (3, 4, -9) (1, 5, 1) 

3x 2y 

z 14 

IFT HER HEY EAS 

2x y 5z 

22 

ONT ONY GIV EYO 

4x 3y 

2z 

18 

URU LIN GER LED 

Which ordered triple is the solution to the system? __________ 

2) System of Equations: (3,9,-2) (-3, 4, 5) (1, 5, -4) (5, 1, 1) 

2x y 4z 

23 

SYS PAPE TEMP RWR 

9x 4z 

7 

ITET HERL UTNA HEO 

x 6y 

4z 

15 

THE RWAY THY OUSE 

Which ordered triple is the solution to the system? __________ 

Place the remaining letters (from left to right) in the blanks. 

____________ ____________ ____________ ____________ ____________ ____________ 

____________ ____________ ____________ ____________ . 

—Juan Ramon Jimenez 

©2010, TESCCC 08/01/10 page 15 of 60

Graphing 22 Systems (pp. 1 of 4) 

Algebra 2 



Getting Started 

Consider the system: 

y 2x 

3 

y 

 

1 2 

x 

2 

For each equation, make a table of 

ordered pairs (x,y). Then graph each line. 

Which ordered pair appears in both 

tables? 

y 2x 

3 

x y 

0 

1 

2 

3 

4 

y 

 

x 

0 

1 

2 

3 

4 

1 2 

x 

y 

2 

What do we call this point, in terms of the system? 

What do we call this point in terms of the graph? 

_______________________ 

_______________________ 

IMPORTANT: 

When the equations in a system can be ___________________, 

then the ___________________ to the system can be described 

using the _____________________ of the _________________ of _____________________. 

 

Why? 

Use the slope and intercept of each linear equation to generate the graphs. Then state the solution to 

the system. 

3 

2 

y 

2 

x 2 

y x7 

3 

1) System: 

2) System: 

y 3x 

1 

y x3 

Solution: 

Solution: 

©2010, TESCCC 08/01/10 page 20 of 60


Algebra 2 



Continue graphing to find the solution to each system. 

3) System: 

y 

y 

 

 

3 

4 

x 

5 7 2 

x 

4) System: 

y 

y 

 

 

1 

2 

3 

2 

x 1 

x 5 

Solution: 

Solution: 

5) System: 

y 

y 

5 

 

1 x 3 

3 

6) System: 

y 

y 

2x 

1 

3 2x 

Solution: 

Solution: 

©2010, TESCCC 08/01/10 page 21 of 60

Algebra 2 




Systems of equations can also be solved using the CALC 5: intersect command on a graphing 

calculator. However, before equations can be entered, you must solve each equation for “y”. 

Sample 

System Equations in “Y=” form Sketch the Graph Table Check Solution 

x 2y 

3x 

4y 

9 

8 

Y 1 = 0.5x + 4.5 

Y 2 = (-3/4)x + 2 

x Y1 Y2 

-4 2.5 5 

-3 3 4.25 

-2 3.5 3.5 

-1 4 2.75 

0 4.5 2 

(-2, 3.5) 

7) System Equations in “Y=” form Sketch the Graph Table Check Solution 

2x 

y 

4x 

y 

5 

12 

Y 1 = 

Y 2 = 

x Y1 Y2 


4x 

7y 

14 

8y 

8 5x 

Y 1 = 

Y 2 = 

x Y1 Y2 


3x 

5y 

30 

6x 

20 10y 

Y 1 = 

Y 2 = 

x Y1 Y2 


3xy 

5 

10 6x 

2y 

Y 1 = 

Y 2 = 

x Y1 Y2 

©2010, TESCCC 08/01/10 page 22 of 60


Algebra 2 



When solving systems 

(such as by graphing), 

three different cases 

can occur. 

What type of lines? 

The lines are… The lines… The lines… 

What type of system? 

What type of solutions? 

______ solutions 

______ solution 

(represented by the 

An _______________ 

of solutions (or, every 

Other Vocabulary 

Sometimes, the 

“__________ ______” 

is used to describe the 

solution to such a 

system. 

Symbol: 

______________.) 

If the two equations in 

the system graph into 

lines which intersect at 

a single point, then they 

are called 

_________________ 

equations 

________ on the line) 

If the two equations in 

the system have the 

same graph, then they 

are called 

_________________ 

equations 

On graph paper or on a calculator, solve each system graphically. Then describe: 

The type of lines—parallel, intersecting, or coincident (coinciding). 

The type of system—inconsistent, consistent independent, or consistent dependent. 

The type of solutions—one solution, no solutions, or infinitely many solutions. 

11) 

y 

y 

2x 

4 

 

1 x 3 

1 

12) 

y 

y 

 

3 

2 

x 1 

3x 

7 

13) 

y 

y 

 

 

2 

5 

2 

3 

x 1 

x 3 

14) 

y 3x 

5 

12x 

4y 

20 

15) 

4x 

2y 

10 

6x 

3y 

12 

16) 

y 0.8x 

2 

5x 

4y 

8 

©2010, TESCCC 08/01/10 page 23 of 60

Systems by Algebraic Methods (pp. 1 of 4) 

Algebra 2 



The “substitution” method can be used to solve systems of equations algebraically. 

x y 11 

7x 4y 

22 

This is the system. 

Solve for a variable in one of the equations. 

In the other equation, replace this variable 

with its equivalent expression. 

Solve (for the numerical value of the 

variable). 

Substitute this answer into another equation 

to find the value of the other variable. 

• Where would you start? 

For each system, without solving, identify which variable in which equation it would be easiest to 

solve for first. Explain why this variable is easiest to solve for. 

4x 

y 3 

7x 

2y 

4 

9x 

11y 

32 

7y 

x 20 

8x 

y 90 

3x 

7y 

112 

3x 

9y 

51 

2x 

4y 

18 

Now solve the systems. Show your work. 

1) 4x y 3 7x 2y 

4 

2) 9x 11y 

32 7y 

x 20 

3) 8x y 90 3x 7y 

112 4) 3x 9y 

51 2x 

4y 

18 

©2010, TESCCC 08/01/10 page 32 of 60

Systems by Algebraic Methods (pp. 2 of 4)) 

Algebra 2 



Sometimes, solving systems with the substitution method can yield strange results… 

Try to solve these systems. 

5) 5x y 2 10x 2y 

9 

6) 3x 9y 

24 

2x 

6y 

16 

When solving a system algebraically… 

If you end up with an equation that is ______________ true, 

4 9 

the system has _____________________ (or, it is _________________________ ) 

or, graphically, the equations’ lines would be ___________________________. 

If you end up with an equation that is _________________ true, 

-24 = -24 

the system has _____________________ (or, it is _________________________ ) 

or, graphically, the equations’ lines would be ___________________________. 

Continue solving these systems algebraically using the substitution method. 

7) 3x y 155 

6x 7y 

5 8) 7x 2y 

3 

2x 

4y 

42 

9) 2x 12y 

40 

x 6y 

19 10) 9x y 6 3x 

5y 

42 

©2010, TESCCC 08/01/10 page 33 of 60

Part B 


Algebra 2 



The “elimination or combination” methods can also be used to solve systems of equations. 

When given a system of equations, you can create other important equations by… 

2x 

5y 

4x 

3y 

19 

24 

12x 

30y 

____ 

12x 

9y 

____ 

24x 

39y 

____ 

_________ 42 

y = 2 

For each equation select a 

constant to multiply by that 

will make the absolute value 

of the coefficient of one of the 

variables equal. 

Multiply both sides of each 

equation by the selected 

constants. 

(Multiplication of equal value 

on both sides of an equation) 

Combine the equations together by addition 

or subtraction. Which eliminates one variable? 

(Addition or subtraction of equal values on 

both sides of an equation) 

In some cases, combining the equations (or, adding them together) eliminates a variable. 

This allows you to solve the system. Try a few on your own. 

2x 

y 11 

a b 19 

5x 

3y 

31 

9x 

2y 

Ex. 

11) 

12) 

13) 

x y 7 

a b 7 

5x 

2y 

4 

x 2y 

17 

7 

3x = 18 

x = 6 

2(6) + y = 11 

y = -1 

• Solution: (6, -1) 

Other times, to eliminate a variable through combination, one equation must first be multiplied by a 

constant. Use this method on the systems below. 

2x 

5y 

19 ( 

2) 

4x 

10y 

38 

3x 

2y 

23 

( 

4) 

__________=____ 

Ex. 

14) 

4x 

3y 

24 

4x 

3y 

24 

2x 8y 

62 

2x 

8y 

62 

2x 

5(2) 19 

2x 

10 19 

2x 

9 

x 4.5 

Solution: (4.5, 2) 

7y 

14 

y 2 

15) 

3x 

2y 

x 5y 

4 

62 

16) 

3x 

2y 

4x 

8y 

10 

8 

©2010, TESCCC 08/01/10 page 34 of 60


Algebra 2 



Finally, you can multiply BOTH equations by a constant, so that when they are combined, a variable 

will be eliminated. Then solve the system. 

Ex. 

8x 

3y 

41 

( 5) 

 

_____________ = 205 

6x 

5y 

39 ( ) 

 

_____________ = -117 

_____________ = 88 

_______ + _______ = _____ _________ = ______ 

Solution: (4, 3) 

MULTIPLY 

the equations by constants so 

that something will “cancel” 

ADD (or COMBINE) 

the equations together 

SOLVE 

the resulting equation 

PLUG 

the value back into an original 

equation to find the other 

variable 

17) 

4x 

5y 

3x 

2y 

70 

18 

18) 

12x 

7y 

8x 

11y 

59 

39 

19) 

10x 

7y 

15x 

8y 

16 

29 

20) 

6x 

7y 

2x 

5y 

47 

21 

©2010, TESCCC 08/01/10 page 35 of 60

Puzzling Systems (pp. 1 of 2) 

Algebra 2 



On a separate piece of paper, solve each system using graphing, substitution, or combination. 

Record the point of intersection, and match it to the letters to complete the coded punch line. 

y 2x98 

D 

y x99 

( _____, _____ ) 

y x6 

N 

x 3y 

( _____, _____ ) 

x 3y 

E 

x2y 

5 

( _____, _____ ) 

x y 6 

R 

3x 

y 4 

( _____, _____ ) 

L 

y x 2 

y 

1 x 2 

4 

( _____, _____ ) 

y x 10 

S 

y 2x 

( _____, _____ ) 

2x 

3y 

11 

U 

x 3y 

7 

( _____, _____ ) 

y x 

4 

I 

x 2y 

( _____, _____ ) 

8x 

4y 

28 

P 

y 2x 

7 

( _____, _____ ) 

y 3x 

14 

H 

2y 

4x 

8 

( _____, _____ ) 

x y 6 

O 

x 3y 

2 

( _____, _____ ) 

y x 1 

A 

y 2x 

2 

( _____, _____ ) 

(1, 100) (-8, -4) (-9, -3) (8, 2) (-10, -20) (3, 4) (-4, -1) (1, 7) 

(2, 8) (3, 1) (4, 2) Infinitely 

many 

solutions 

(3, 1) (1, 7) 

©2010, TESCCC 08/01/10 page 47 of 60

Puzzling Systems (pp. 2 of 2) 

Algebra 2 



On a separate piece of paper, solve each system using graphing, substitution, or combination. 

Record the point of intersection, and match it to the letters to complete the coded punch line. 

y 3x 

3 

H 

6x 

3y 

6 

( _____, _____ ) 

11x 

2y 

1 

C 

9x 

3y 

24 

( _____, _____ ) 

Don’t argue with a dummy! 

People watching may not 

be able _____. 

2x 

2 2y 

L 

2x 

y 2 

( ____, _____ ) 

4x 

8y 

16 

O 

3x 

4y 

2 

( _____, ____ ) 

D 

y 2x 

6 

y 

1 x 2 

( _____, _____ ) 

4 

3x 

y 3 

R 

6x 

3y 

3 

( _____, ____ ) 

y x 

12 

E 

y x 4 

( _____, _____ ) 

3 

y x 

2 

T 

y 2x 

7 

( _____, ____ ) 

x y 6 

F 

x y 10 

( _____, _____ ) 

4x 

6y 

3 

N 

2x 

3y 

10 

( _____, ____ ) 

2x 

6y 

26 

I 

y x 3 

( _____, _____ ) 

(-2, -3) (2, 1) (-2, -3) (8, 4) (-3, 4) (-3,4) (-2, -3) (1, 0) (8, 4) 

(4, 2) (1, 4) (8, 2) (8, 2) (8, 4) (4, 9) (8, 4) No Sol. (1, -5) (8, 4) 

©2010, TESCCC 08/01/10 page 48 of 60

Systems Situations (pp. 1 of 3) 

Algebra 2 



For each situation, define variables and set up a system of equations. If you have trouble at first, try 

one of these strategies to help: 1) List possibilities, 2) Trial and error, 3) Grouping like numbers. 

1. Copies. A teacher has to make 159 copies of a review packet for the students in her classes. 

When she arrives in the workroom, she starts using both copy machines (call them X and Y). 

During the copying, machine X runs out of paper, so it only printed half as many packets as 

copier Y. How many copies does each machine print? 

Variables: 

x = Number of copies made by Machine X 

System: 

y = Number of copies made by Machine Y 

2. Fast Food. A bunch of friends went to the Snack Shack for lunch. The first family ordered 4 

hamburgers and 4 orders of fries for $9.00. The next family ordered only 1 hamburger and 2 

orders of fries for $3. How much would each item cost individually? 

Variables: 

System: 

= 4h + 4f = 9 

= h + 2f = 3 

3. Exercise. Several times a week, Chuck goes to the gym to run and swim. When running, Chuck 

burns 35 calories per minute, and when he swims he burns 30 calories per minute. He has 

found a way to burn 730 calories after exercising for a total of 23 minutes. How long does Chuck 

spend at each activity? 

Variables: 

= 

System: 

= 

4. Coins. Donald has a bunch of nickels and dimes in his piggy bank. If there are 100 coins in the 

bank that make a total of $6.60 in change, how many of each type of coin does Donald have? 

Variables: 

= 

System: 

= 

©2010, TESCCC 08/01/10 page 52 of 60


Algebra 2 



5. Golf. To play golf at Blackhawk Range, golfers must first pay a $75 membership fee, and must 

still pay $7 for every round of golf. At the Royal Estates Country Club Course, golfers only have 

to pay $5 per round, but their membership dues are $185. After how many rounds of golf would 

the total amount paid by a golfer be the same at both golf courses? 

Variables: 

= 

System: 

= 

6. Discs. There are 43 discs in Josh’s entertainment center. However, some of them are music 

CD’s and some of them are video DVD’s. If the number of music CD’s in Josh’s collection is 5 

less than three times the number of DVD’s, how many of each type does he own? 

Variables: 

= 

System: 

= 

7. Salesman. Carmen sells cars at a dealership that pays her a monthly salary of $2500 plus a 

commission of $125 for every car she sells. A competing dealership has offered her a job that 

pays a greater monthly salary of $2700, but only gives $100 commission on each car sold. How 

many cars would Carmen have to sell in a month to get the same pay from either dealership? 

Variables: 

= 

System: 

= 

8. Circus. There are 18 performers in Barnum’s Famous Flea Circus. However, none of them are 

actually fleas—they are either spiders (with 8 legs) or bugs (with 6 legs). Mr. Barnum wants to 

provide every performer with tiny new shoes for the ends of each of their many legs. If this 

requires a total of 130 shoes, how many spiders and bugs are in the circus? 

Variables: 

= 

System: 

= 

©2010, TESCCC 08/01/10 page 53 of 60


Algebra 2 



9. Amused. Several families are in line to enter an amusement park. The first family pays $54 for 

two adult tickets and two child tickets. The second family pays $105 for 3 adults and 5 children. 

What are the ticket prices for adults and for children? 

Variables: 

= 

System: 

= 

10. Age. Samantha has an older brother. Right now, he is twice her age. But 8 years ago, her 

brother was 3 times her age. How old is each person now? 

Variables: 

= 

System: 

= 

11. Health. Sherri is considering joining a health club. To be a member of Roman’s Gym, she would 

have to pay a sign-up fee of $49.99 and then $25 per month. Another club, Athens Fitness 

Center, charges less per month (only $18) but has a $126.99 sign-up fee. After how many 

months of membership would the total paid to each club be the same? 

Variables: 

= 

System: 

= 

12. More Discs. After cleaning house, Josh took a box of 29 used discs to sell at the flea market. 

Some were computer games (which he sold for $2 each), some were music CD’s (which he sold 

for $3 each), and the rest were video DVD’s (which he sold for $4 each). If Josh sold everything 

(which included twice as many DVD’s as computer games), and he made a total of $93, how 

many of each type of disc did he sell? 

Variables: 

= 

System: 

= 

= 

©2010, TESCCC 08/01/10 page 54 of 60

Situation Problems Involving Systems 

Algebra 2 



1. The band booster club is selling coffee and hot chocolate at the soccer games. Both drinks are 

sold in the same paper cup. They sold $272.25 worth of hot drinks and used a total of 275 

cups at the last game. If hot chocolate sells for $1.25 and coffee sells for $.75, how many of 

each type of hot drinks did they sell? 

2. Susie is going to purchase a new freezer. One brand costs $1800 and estimates $60 per 

month to operate. A more expensive brand costs $2600, but only costs $40 per month to 

operate. 

a. Write an equation representing the cost of each freezer in terms of months owned. 

b. Find the total cost of each after 1 year, 2 years, and 3 years. 

c. Plot the graph of each on the same axes. 

d. Find the break-even point for the two freezers. When does this occur? What is this 

cost? 

e. If Susie plans on keeping this freezer for five years, which freezer would be the most 

economical? Explain. 

3. Treebolt High School is running an inventory of graphing calculators. They have two types, TI- 

83 + and TI-83 + Silver Edition. Records show they purchased 350 calculators for a total of 

$37,750. If TI-83 + costs $95 and TI-83 + Silver Edition costs $125, how many of each type of 

calculator should the school have in inventory? 

4. One brand of car with a regular engine costs $21,500 to purchase, and 32 cents per mile to 

operate. The same car with a fuel-injection engine costs $22,500 to purchase and only 29 

cents per mile to operate. 

a. Write an equation for each particular car comparing total cost in terms of miles 

operated. 

b. Calculate the cost of operating each vehicle for 1,000, 10,000, and 50,000 miles. 

c. Plot the graph of each on the same axes. 

d. Find the break-even point for the two vehicles. At what mileage does this occur? What 

is this cost? 

e. If Trey switches cars after 40,000 miles, which car would be the most economical? 

Explain. 

©2010, TESCCC 08/01/10 page 44 of 60

Solving Systems with Technology and Matrices (pp. 1 of 3) 

Algebra 2 



How can we solve problems requiring more variables and more equations using technology? First, 

you must make sure that the system is in “standard form.” This means that the variable terms are 

lined up on the left side of the equations, and the constants are on the right. 

Sample System: 

x 2y 

z 9 Looks good 

2x 

y 3z 

Add 3z to both sides 

3x 

1 4y 

5z 

Add 1, -4y, and -5z to both sides 

Standard Form: 

x 2y 

z 9 

2x 

y 3z 

0 

3x 

4y 

5z 

1 

Using Inverse Matrices 

With this method, think of the system as… 

33 matrix 

of 

coefficients 

 

31 

matrix of 

variables 

= 

31 

matrix of 

constants 

In this case: 

1 

 

 

2 

 

3 

2 1 x 

9 

1 3 

 

 

 

 

 

 

y 

 

0 

 

4 5 

 

z 

 

1 

Enter the coefficients in one matrix (here, [A]): 

Using Augmented Matrices 

With this method, think of the system as an 

augmented 34 matrix. 

1 

2 1 9 

 

 

 

2 1 3 0 

 

 

3 4 5 1 

The first three 

columns are 

coefficients 

The 

variables 

are not 

shown 

The last 

column 

holds the 

constants 

Enter this augmented matrix into the calculator: 

Enter the constants in another (here, [C]): 

Compute the solutions using [A] -1 [C]: 

From the MATRIX 

find the command: 

MATH menu, scroll down to 

rref( 

This places a matrix in “reduced row echelon” 

form. 

So, the solutions are: 

x 

 

 

y 

 

 

z 

2 

 

 

 

5 

 

 

3 

Here, solutions appear in the last column. 

So, the solutions are: 

x 

2 

 

 

y 

 

 

 

5 

 

 

z 

 

3 

©2010, TESCCC 08/01/10 page 14 of 40

Special Cases 


Algebra 2 



Inverse Matrices 

Augmented Matrices 

1 

0 0 5 

Infinite number of 

Infinite number of 

 

 

ERR: SINGULAR MAT 

0 1 0 2 

solutions 

solutions 

 

 

 

0 0 0 0 

1 

0 0 5 

 

 

No solutions ERR: SINGULAR MAT No solutions 

0 1 0 2 

 

 

 

0 0 0 1 

Since the inverse matrices do not distinguish between infinite and no solutions, another method must 

be used to determine which one is the correct answer. 

Examples: 

1. 

x 2y 

2z 

7 

4z 

3x 

1 

y z 6 



Standard 

form 

Standard 

form 

Written as 

Matrix 

Equation 

Written as 

Augmented 

Matrix 

Matrix 

Solution 

Matrix 

Solution 

Values of 

x,y,z 

Values of 

x,y,z 

©2010, TESCCC 08/01/10 page 15 of 40


Algebra 2 



2. 

x 2y 

2z 

7 

4z 

3y 

1 

x 6 z 



Standard 

form 

Standard 

form 

Written as 

Matrix 

Equation 

Written as 

Augmented 

Matrix 

Matrix 

Solution 

Matrix 

Solution 

Values of 

x,y,z 

Values of 

x,y,z 

©2010, TESCCC 08/01/10 page 16 of 40

Learning About Lingos 

Algebra 2 



Solve each system. 

Find the code letter for 

the answers and fill in 

the blanks below to 

answer the questions. 

A B C D E F G H 

2,1,3 3,1, 12 

1 

1, , 1 

3 

 

 

6,6,6 1, 2, 2 5,7, 1 

1, 6, 6 1, 2,3 

I L M N O P R S W 

1 

1 1 1 

1 1 

1, 1, 6 1, ,1 

3 

1,3, 2 ,0,6 

 

2 

, ,2 

23 

7,2,0 6,6,3 

, ,1 

 

23 

6, 2,5 

 

1) 

2xy 3z 

1 

3xy 2z 

4 

x2y z 

5 

2) 

3x4y z 

1 

x3y z 

2 

2xy 2z 

1 

3) 

4x 5y z 

8 

4x3y 2z 

10 

2xy z 

5 

4) 

3x3y z 

3 

2x3y 2z 

3 

x6y 3z 

0 

5) 

4xy 62z 

3x 2y 53z 

x y z 

1 

6) 

2x z13y 

2x 2z 6y 

5x4y 3z 

6 

7) 

2x 6y z 

5 

x y z10 

x 8y 2z 

0 

8) 

2xy 3z 

19 

3x2y 3z 

19 

x4y 2z 

9 

9) 

4x 6y 3z 

2x5z 3y 

5 

9y 4z 6x4 

10) 

2x 7 y 4z 

2x y 2z 

x 3z 222y 

11) 

2x y z 

12 

3xy 4z 

36 

xy z 

18 

Did you know that more than … 

8 million people speak _____ _____ _____ _____ _____ _____? 

9 8 3 11 6 8 

14 million people speak _____ _____ _____ _____ _____ _____ _____? 

9 7 2 6 8 4 8 

5 million people speak _____ _____ _____ _____ _____ _____ _____? 

10 4 5 1 8 9 6 

©2010, TESCCC 08/01/10 page 18 of 40

Mixing It Up (pp. 1 of 5) 

Algebra 2 



WARM UP 

Compute answers to each of the following “mixture” problems. 

A) Florence took a road trip to the beach, which was several hundred 

miles away. As she left, she had to drive 36 mph through the city 

for about half an hour. Once she hit the highway, she drove 68 

mph for 3.5 hours. Then, as she approached the beach, the traffic 

was so heavy she could only go 20 mph for the last 30 minutes. 

 

 

 

How many hours was Florence on the road? 

How many miles did Florence travel? 

Over the course of her trip, what was Florence’s average speed? 

B) When you buy hydrogen peroxide, it is really a solution diluted 

with water. One bottle comes in a 2% solution, and another 

comes in 5% strength. Suppose you mix 40 ml of the 2% solution 

with 80 ml of the 5% solution. 

 

How many milliliters of the mixture is actually pure hydrogen 

peroxide? 

 

What percent of this mixture is hydrogen peroxide? 

C) Donna always gets really high grades on tests and quizzes (95’s 

every time), but she only does homework half the time (so her 

homework grade is a 50). 

 

Donna’s teacher has a policy where tests and quizzes make 

up only 30% of students’ grades and homework makes up 

the other 70%. Under this system, what would Donna’s class 

grade be? 

 

A teacher down the hall uses a different policy, where tests and quizzes are 60% of the 

grade, and homework is only 40%. What would Donna’s grade be if she had this teacher? 

©2010, TESCCC 08/01/10 page 24 of 40


Algebra 2 



For each situation, define variables and set up a system of equations that relates them. Then, solve 

the system using matrices. 

1) When traveling to see relatives, Duke drives part of the trip on the highway at 60 mph and the 

rest through cities at 40 mph. The 270-mile trip takes him a total of 5 hours. How much of this 

time does he spend on the highway? How much time through cities? 

Solutions: 

= 

System: 

= 

Matrix Equation: 

2) On the return trip, Duke encounters some road construction that slows him down. While he is 

still able to go 60 mph on the highway and 40 mph through cities, he can only go 20 mph past 

the construction. He spends twice the time passing the construction as he does on the highway, 

so this time the 270-mile trip takes 7.8 hours. How much of this time does he spend on the 

highway, in the city, and through road construction? 

Solutions: 

= 

= 

System: 

= 


©2010, TESCCC 08/01/10 page 25 of 40


Algebra 2 



Continue setting up and solving systems for these situations. 

3) In the school’s chemistry lab, a certain acid is stored in two different dilutions. One bottle 

contains a weaker solution that is only 25% acid, but another has a stronger solution of 65% 

acid. Students must pour some from each container to make 60 ml of a solution that is 50% 

acid. How much from each container should be used? 

Solutions: 

= 

System: 

= 


4) When Ebenezer won a $250,000 lottery jackpot, everyone started asking for donations. 

His brother Bob said, His sister Sally said, And his friend Fred said, 

“I need a loan! I can pay 

you back in a year. I’ll 

even pay you back with 

4% interest!” 

“I need $20,000 more than Bob 

does, but I want to start my own 

business. And you’ll earn more 

than 4% on this sweet deal.” 

“No way. You need to put 

your money in the stock 

market. It’s doing great right 

now.” 

Ebenezer split all his money among these three investments. Bob repaid his loan and interest, 

and the investment in Sally’s business came back with 7% profit. Even though the money he 

put in the stock market lost 8.2%, Ebenezer still came out with an extra $5,830. 

How much money did he place into each of these investments? 

Solutions: 

= 

= 

System: 

= 


©2010, TESCCC 08/01/10 page 26 of 40


Algebra 2 




5) When extreme athletes compete at a motorcycle jumping competition, judges give them scores 

on a scale of 0-10 in three categories: use of fundamentals, degree of difficulty, and overall 

style. However, these categories are not weighted equally in determining the final score. 

Funda- Difficulty 

score 

Final 

Style 

mentals 

D. Howser 7.4 8.2 8.8 8.01 

R. Finn 7.4 7.8 9.6 8.23 

J. Hoban 8.0 7.6 8.6 8.15 

= 

= 

The table shows how the category 

scores for three riders determined their 

final rating. 

By what percent is each category 

weighted? 

Solutions: 

System: 

= 


6) At the lake, Curtis caught three fish. He wanted to weigh 

the fish individually, but his scale was broken--It could only 

read weights between 5 and 10 pounds. So he weighed 

the large and medium fish together, and got 7.8 pounds. 

The large and small fish weighed 7.0 pounds, and the 

small and medium fish were 5.6 pounds. 

How much did each fish weigh individually? What was the total weight of all three fish? 

Solutions: 

= 

= 

System: 

= 


©2010, TESCCC 08/01/10 page 27 of 40


Algebra 2 




7) On the way home from school, a family enters the drive-through at a fast food restaurant. 

Mom: “I need 3 root beers, 2 junior burgers and one deluxe salad.” 

Clerk: “That will be $10.62. Please pull around.” 

Lucy: “Mom, I don’t want a burger. I’m a vegetarian now!” 

Mom: “OK—sorry. Can I get 3 root beers, 1 junior burger and 2 deluxe salads, instead?” 

Clerk: “No problem, ma’am. That brings your total to $11.22.” 

Mom: “Oh no. I’ve only got $11. Could you take off one of the root beers?” 

Clerk “Yes. Two root beers, 1 junior burger, and 2 salads. That’s $10.03.” 

What is the cost for each item individually? 

Solutions: 

= 

= 

= 

System: 


8) A box in the textbook warehouse has the given label. It contains 

some Algebra II textbooks (each costs $84 and weighs 5.2 

pounds), some Chemistry books (each costs $90 and weighs 4.4 

pounds), and some Literature books (each costs $75 and weighs 6 

pounds). How many of each type of book are in the box? 

= 

Smart School Publishing 

Quantity: 17 books 

Weight: 84.4 pounds 

Value: $1,449 

Solutions: 

= 

= 

System: 


©2010, TESCCC 08/01/10 page 28 of 40

Quadratic Quest (pp. 1 of 2) 

Algebra 2 



2 

Quadratic functions have the general equation y ax 

If coordinates for three points on the parabola are known, 

then the equation can be found using a 33 system of 

equations to find a, b, and c. 

For example, the given quadratic function passes through 

Q(-1, 1.375), R(2.5, 4) and D(3.5, 2.5). This means… 

2 

ax + b x + c = y 

For Q… a(___) 2 + b(___) + c = ____ 

For R… a(___) 2 + b(___) + c = ____ 

For D… a(___) 2 + b(___) + c = ____ 

bx c 

, where a 0 . 

This system can be written in matrix form and solved to generate the quadratic function. 

Matrices Solution Quadratic Function 

 

a 

a 

y = 

 

 

 

 

 

 

 

 

b 

 

 

b 

 

 

 

Check this on a calculator! 

 

 

 

c 

 

 

 

c 

Use this strategy to solve more problems involving quadratic functions. 

1) The parabola shown has its vertex at V(0.5, -2.6). A point on 

the left side of the parabola is located at L(-3, 2.3). 

What are the coordinates for R, the corresponding point 

on the right side? 

Set up a system to find the constants a, b, and c. 

2 

ax + b x + c = y 

For L… a(___) 2 + b(___) + c = ____ 

For V… a(___) 2 + b(___) + c = ____ 

For R… a(___) 2 + b(___) + c = ____ 

 

 

 

 

 

Matrices Solution Quadratic Function 

a 

a 

y = 

 

 

 

 

 

 

b 

 

 

b 

 

 

 

Check this on a calculator! 

 

 

c 

 

 

 

c 

From the graph, it looks like the function has x-intercepts at approximately -2 and 3. Find these 

values, exactly. 

©2010, TESCCC 08/01/10 page 31 of 40

Quadratic Quest (pp. 2 of 2) 

Algebra 2 



2) Bart throws a water balloon off the top of a 40-ft building. 

From surveillance photographs, it was determined that, after 

he released the balloon, its height at 0.8 seconds was 41.76 

feet, and it was 23.04 feet high after 1.6 seconds. 

The height (h, in feet) of the water balloon is a quadratic 

function of time (x) in seconds. Find this function in the 

2 

form h( x) 

ax bx c . 

Time Height 

(sec) (ft) 

0 40 

0.8 41.76 

1.6 23.04 

? ? 

? ? 

 

 

Graph the function in a calculator. After precisely how many seconds did the water balloon 

hit the ground? 

How high did Bart throw the balloon above the height of the building? 

3) The journalism department has put together a magazine that features students’ poems and 

short stories as well as pictures of student artwork. They plan on selling the collections as a 

fundraiser, but are unsure what to charge. They can generate more revenue if they charge 

more for the magazine—but if it’s too expensive, no one will buy it (and they’ll lose money). By 

surveying students in English classes, they determine that charging $4 for the books will 

generate $712 in revenue, and charging $6 each will create $852 in sales; but at $11 apiece, 

they will only bring in $572. 

The revenue (R, in dollars) earned from the sale of the magazines is a quadratic function of 

2 

their individual cost (x). Find this function in the form R( x) 

ax bx c . 

 

Using a calculator, determine how much money they could make it they charged $12 for 

each magazine. 

 

About how much should they charge to earn the most money? According to the function, 

what’s the most they could make? 

©2010, TESCCC 08/01/10 page 32 of 40

Graphing Inequalities (pp. 1 of 3) 

Algebra 2 



The symbol (>,


Algebra 2 



The solution to a system of inequalities is the region that would be included by every inequality. 

To graph 1. Lightly shade each individual inequality. (This shows your “work.”) 

a solution: 2. Heavily shade the region where they overlap. (This is the solution.) 

6) 

y 

y 

3x 

5 

1 x 2 

2 

7) 

x y 3 

y 2x 

3 

y 3 

8) 

3x 2y 

12 

9) 

y 2x 

1 

x 0 

x 

y 4 

y 2x5 

x 2 

y 3 

10) What system of inequalities could be used to create this graph and 

solution? 

©2010, TESCCC 08/01/10 page 29 of 61


Algebra 2 



Graph these systems for additional practice. 

11) 

5x 

2y 

10 

y x 

4 

12) 

y x 2 

y 

 

1 

2 

x 2y 

x 1 

10 

13) 

x 2 

y 3 

2x 

y 3 

y 2x 

4 

14) 

y x 

3 

y 4x 

1 

3x 

2y 

8 

©2010, TESCCC 08/01/10 page 30 of 61

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