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SolvingSystems of Equations
11th Grade7-Day Unit Plan
Tools Used:
TI-83 graphing calculator (teacher)Casio graphing calculator (teacher)TV connection for Casio (teacher)Set of Casio calculators (students)
By: Nicole M. McCoy
2
Objectives of Unit:
ÿ The students will recognize the properties of systems of equationsÿ The students will discover four different methods to solving systems of equationsÿ The students will be able to choose the “easiest” method when solving a system
Standards addressed throughout lesson:
NCTM Standards
Numbers & OperationsAlgebraProblem SolvingCommunicationConnectionsRepresentation
NYS Key Ideas
Key Idea 2- Number and NumerationKey Idea 3 – OperationsKey Idea 4 – Modeling/Multiple RepresentationKey Idea 7 – Patterns/Functions
Textbook Information:
Publisher – Scott Foresman Addison WesleyTitle – The University of Chicago School Mathematics ProjectAuthors – Senk, Thompson, Viktora, Usiskin, Abbel, Levin, Weinhold, Rubenstein, Jaskowiak, Flanders, Jakucyn, PillsburyChapter/pages Chapter 5/pages 279 - 311Copyright 1998
3
Unit Overview
Day 1Solving Systems of Equations Graphically
Day 2Solving Systems of Equations Algebraically – Using
Substitution
Day 3Solving Systems of Equations Algebraically – Using Linear
Combinations
Day 4Review of Linear Combinations; Preview to Matrices
Day 5Solving Systems of Equations Using Matrices on the Casio
Day 6Solving Systems of Equations using any method – word
problems
Day 7More word problems and quiz
4
Day 1
Lesson: Solving systems of equations graphically
Objectiveso Students will recognize properties of systems of equationso Students will estimate solutions to systems by graphing
Extra Materialso None
Opening Activity
Student Bellwork:The students have 5 minutes to complete the following problem. The students will then pass the paper to adifferent student and they will correct each others papers. The teacher will collect the papers after wecorrect them as a class.
Have the students graph the following on separate graphs (provided by teacher):
4
2
532
=
=
-=
xy
xy
xy
This will determine who knows how to use their graphing calculator and who needs more help/practice.
Main Activity
As a class, we will discuss how to find the solution to a system of linear equations using the graphs of thelines. NOTE – This method can only be used if solving for 1 or 2 variables due to the limited powers (no 3-Dcapabilities on the graphing calculators) This will be mentioned to the students early in the lesson. The studentswill find the solution(s) to the following systems while working in groups. Then, each group will present onesystem and show how they arrived at the solution(s).
1.ÓÌÏ
+-=
-=
52
7
xy
xy2.
ÓÌÏ
+=
+=
1
133
xy
xy3.
ÔÓ
ÔÌ
Ï
-=-
+=
22
12
1
yx
xy
4.ÓÌÏ
+=
=-
69
129
xy
yx5.
ÓÌÏ
+=
=+
26
73
xy
xy6.
ÓÌÏ
+-=
=
13
5
xy
xy
Closing Activity
Have the students write a brief sentence or two answering this question:In example 4, there was no solution because the lines were parallel. Give a situation where there would be aninfinite amount of solutions to a system of linear equations. Think about the parallel lines example.
Homework
Page 283 # 10-12, 15
5
Day 2
Lesson: Solving systems of equations algebraically using substitution
Objectiveso Students will solve 2x2 and 3x3 systems using substitution o Students will recognize properties of systems of equations
Extra Materialso Class set of Lesson Master 5-3 B
Opening Activity
Student Bellwork:The students have 5 minutes to complete the following problem. The students will then pass the paper to adifferent student and they will correct each others papers. The teacher will collect the papers after wecorrect them as a class.
Have students solve the following:If y = -1, solve the following equations for x:
1. x = -2y + 42. 4x + 3y = 93. 28 yx +=
Answers:1. x = 62. x = 33. x = 7
Main Activity
Students will learn how to solve systems of equations using the substitution method and Lesson Master 5-3 B.Together we will discuss how to use substitution to solve for the variables.
The teacher will put the following example on the board:5 = 2 + 34 + 1 = 5
Ask the students how to change this from two different equations to one equation that is equivalent.Answer: 2 + 3 = 4 + 1; because they both equal 5, they must both be equal to each other.
Discuss when and why we use substitution.- It is used when we have more than 1 unknown (variable) and one of the equations can be solved for
1 variable. Then we substitute that variable’s value in terms of the other variable(s) into the otherequation.
Discuss how to use substitution.- This is when we, as a class, will work through the odd numbered problems on the Lesson
Master 5-3 B. The students will solve 2x2 and 3x3 systems using substitution.
6
Closing Activity
On a half sheet of paper students will comment on the substitution method for solving systems of equations.They will include:
- When and why we use substitution- How to use substitution- If they prefer graphing the two lines or solving algebraically and why
This will be the “Ticket out of Class” and each student needs to hand the teacher a paper to exit the room. (NOTE:No late passes will be allowed if student refuses!)
Homework
Finish worksheet
7
Lesson Master 5-3B
In 1-8, use substitution to solve the system. Then check.
1.ÓÌÏ
+-=
-=
52
7
xy
xy2.
ÓÌÏ
+=
+=
1
133
xy
xy
3.ÓÌÏ
=-
=-
11621
123
nm
nm4.
ÓÌÏ
-=
=
yx
xy
4
4
5.ÓÌÏ
=-
=+
9905.15.7
781.25.
xy
yx6.
ÔÓ
ÔÌ
Ï
-=
+=
-=-+
ac
ab
cba
4
3
26364
7.ÔÓ
ÔÌ
Ï
+=
+-=
=+
1
1
10
xy
xz
zxy
8.ÔÓ
ÔÌ
Ï
-=-
+=
22
12
1
yx
xy
8
Lesson Master 5-3B (Answer Key)
In 1-8, use substitution to solve the system. Then check.
1.ÓÌÏ
+-=
-=
52
7
xy
xy2.
ÓÌÏ
+=
+=
1
133
xy
xy
x=4; y=-3 x=-6; y=-5
3.ÓÌÏ
=-
=-
11621
123
nm
nm4.
ÓÌÏ
-=
=
yx
xy
4
4
m=2/3; n=1/2 x=-4; y=1x=4; y=-1
5.ÓÌÏ
=-
=+
9905.15.7
781.25.
xy
yx6.
ÔÓ
ÔÌ
Ï
-=
+=
-=-+
ac
ab
cba
4
3
26364
x=240; y=180 a=-2; b=1; c=8
7.ÔÓ
ÔÌ
Ï
+=
+-=
=+
1
1
10
xy
xz
zxy
8.ÔÓ
ÔÌ
Ï
-=-
+=
22
12
1
yx
xy
x=3; y=4; z=-2 infinitely many solutionsx=-3; y=-4; z=2
9
Day 3
Lesson: Solving systems of equations algebraically using linear combinations
Objectiveso Students will solve 2x2 and 3x3 systems using linear combinations o Students will recognize properties of systems of equations
Extra Materialso Class set of Lesson Master 5-4 B
Opening Activity
Student Bellwork:The students have 5 minutes to complete the following problem. The students will then pass the paper to adifferent student and they will correct each others papers. The teacher will collect the papers after wecorrect them as a class.
Have students solve the following:y = 2x + 34x + 3y = 29
Answers:x = 2y = 7
Main Activity
Students will learn how to solve systems of equations using the linear combinations method and LessonMaster 5-4 B. Together we will discuss how to use linear combinations to solve for the variables.The teacher will put this example on the board:
4x – 6y = 32x + 12y = -6
Ask the following questions:1. How would we get the coefficients of the x’s to be the same number with opposite signs? Answer:
Multiply the 2x by -22. How would we get the coefficients of the y’s to be the same number with opposite signs? Answer:
Multiply the 6y by 2
Discuss when and why we use linear combinations.- It is used when we are not able to solve for one variable easily (or at all) and/or one equations variable
is a multiple of the others’.
Discuss how to use linear combinations.- This is when, as a class, we will do the problems together on Lesson Master 5-4 B. Students will solve
2x2 and 3x3 systems using linear combinations.
Closing Activity
On a half sheet of paper students will discuss which algebraic method of solving the systems of equationsthey would use to solve the following problems and why.1. 6x + 12y = 5 (substitution) 2. x + y = 9 (linear combinations)
y = 2 – 10x 2x – y = 2
10
3. 2x + 3y + z = 13 (linear combinations) 4. y = 3x (substitution)5x – 2y – 4z = 7 xy = 484x + 5y + 3z = 25
This will be the “Ticket out of Class” and each student needs to hand the teacher a paper to exit the room. (NOTE:No late passes will be allowed if student refuses!)
Homework
Finish worksheet 5-4 B
11
Lesson Master 5-4B
In 1-8, use linear combinations to solve the system. Then check.
1.ÓÌÏ
-=+
-=+
1522
124
yx
yx2.
ÓÌÏ
=-
=+
1.225
6.234
yx
yx
3.ÔÓ
ÔÌ
Ï
=+-
-=-+
-=-+
184
1522
2152
cba
cba
cba
4.ÓÌÏ
-=-
-=-
45.2
1628
nm
nm
5.ÔÓ
ÔÌÏ
=+
=-
48226
52351222
22
yx
yx6.
ÓÌÏ
-=+
-=+
20108
1454
yx
yx
7.
ÔÔÓ
ÔÔÌ
Ï
=+
-=-
1442
1
84
1
yx
yx8.
ÔÓ
ÔÌ
Ï
-=++
-=++
=-+
6222
723
139
fed
fed
fed
12
Lesson Master 5-4B Answer Key
In 1-8, use linear combinations to solve the system. Then check.
1.ÓÌÏ
-=+
-=+
1522
124
yx
yx2.
ÓÌÏ
=-
=+
1.225
6.234
yx
yx
x=-1.5; y=-6 x=.5; y=.2
3.ÔÓ
ÔÌ
Ï
=+-
-=-+
-=-+
184
1522
2152
cba
cba
cba
4.ÓÌÏ
-=-
-=-
45.2
1628
nm
nm
a=-1; b=-4; c=3 infinitely many solutions
5.ÔÓ
ÔÌÏ
=+
=-
48226
52351222
22
yx
yx6.
ÓÌÏ
-=+
-=+
20108
1454
yx
yx
x=8; y=7 no solutionx=8; y=-7x=-8; y=7x=-8; y=-7
7.
ÔÔÓ
ÔÔÌ
Ï
=+
-=-
1442
1
84
1
yx
yx8.
ÔÓ
ÔÌ
Ï
-=++
-=++
=-+
6222
723
139
fed
fed
fed
x=-12; y=5 d=0; e=1; f=-4
13
Day 4
Lesson: Review of Linear Combinations and Solving systems of equations algebraically using matrices ongraphing calculator
Objectiveso Students will solve 2x2 and 3x3 systems using matrices on graphing calculatoro Students will recognize properties of systems of equations
Extra Materialso Students worksheets from previous class day (Lesson Master 5-4 B)
Opening Activity
Student Bellwork:The students have 5 minutes to complete the following problem. The students will then pass the paper to adifferent student and they will correct each others papers. The teacher will collect the papers after wecorrect them as a class.
Have students solve the following:Solve the following system of equations using linear combinations:
2x – y = 4 Answer: x = 4-x + 3y = 8 y = 4
Main Activity
Students will review how to use the linear combination method to solve systems of equations. As a class,we will go over the homework problems that caused the students difficulty.
After all homework questions have been answered and the students have done a 3x3 system, we willquickly set up the matrix method on the calculators.
STEPS: Turn calculator on to the main menu. Go to EQUA and hit EXE. This is a simultaneous process, so hit F1for simultaneous. Next, we need to enter the number of unknowns. For a 3x3 we will have 3 unknowns(variables). Enter in all of the coefficients and the constants as prompted on the screen. After all the numbers areentered, hit F1 to solve. This will give the answer matrix to the 3x3 system we entered. Ask the students whichway they prefer. I guarantee it will be the matrix method.
Closing Activity
Have the students try another system from their homework on the calculator. Have them do as many astime will allow.
Discuss with them quickly that this will not however be accepted as full credit work on an exam. There issome work involved and we will discuss that tomorrow.
14
Day 5
Lesson: Solving systems of equations algebraically using matrices on the calculator and why it works
Objectiveso Students will solve 2x2 and 3x3 systems using matrices on the calculator o Students will recognize properties of systems of equationso Students will understand the steps involved to solve systems using the matrix method
Extra Materialso Class set of worksheets
Opening Activity
Student Bellwork:The students have 5 minutes to complete the following problem. The students will then pass the paper to adifferent student and they will correct each others papers. The teacher will collect the papers after wecorrect them as a class.
Have students solve the following system using the matrix method:2x – 2y + 4z = 7 Answer: x = -33-4x + 2y – 3z = 14 y = -126.5x + 4y -12z = 1 z = -45
Main Activity
Have the students learn how the matrix method works without actually showing them how to use a matrix.We will as a class go through the steps to “solving” a system using the matrix method.
Step 1: Change system of equations to matrix equation:
ÔÓ
ÔÌ
Ï
=+
=+
=+
1 12z-4y x
14 3z2y 4x -
7 4z 2y 2x
This system becomes the following matrix
coefficient matrix constant matrix
˙˙˙
˚
˘
ÍÍÍ
Î
È
-
--
-
1241
324
422
˙˙˙
˚
˘
ÍÍÍ
Î
È
z
y
x
=
˙˙˙
˚
˘
ÍÍÍ
Î
È
1
14
7
variable matrix
Step 2: Label the matrices:
˙˙˙
˚
˘
ÍÍÍ
Î
È
-
--
-
1241
324
422
˙˙˙
˚
˘
ÍÍÍ
Î
È
z
y
x
=
˙˙˙
˚
˘
ÍÍÍ
Î
È
1
14
7
A B
15
Step 3: Multiply [ ] 1-A x [ ]A so that all we have on the left side of the equation is the variable matrix. And,
because we have an equation, we need to multiply [ ] 1-A x [ ]B on the right side of the equation.
So we now have:
[ ] 1-A x [ ]A
˙˙˙
˚
˘
ÍÍÍ
Î
È
z
y
x
= [ ] 1-A x [ ]B
Step 4: [ ] 1-A x [ ]A cancel out to become the identity matrix (essentially 1) and so we are now left with:
˙˙˙
˚
˘
ÍÍÍ
Î
È
z
y
x
= [ ] 1-A x [ ]B , where [ ] 1-A x [ ]B is the answer matrix.
This is enough work to show for full credit. Now, the students will put numbers into the calculator to get theanswer matrix. They will work on the worksheets in pairs and solve for the variables, showing all of their work.
Closing Activity
Explain in your own words why [ ] 1-A x [ ]A cancels out (becomes 1). Give an example using numbers.
Example Answer: 12
12 =⋅ , where
2
1 is the inverse and 1 is the identity.
Homework
Finish worksheet
16
Name _____________________________________ Date ________________
Solving Systems of Equations using Matrices
Directions: Solve the following systems of equations using the matrix method on your calculator.
1) 3x – y + 4z = -17 1) Number of unknowns ________4x + 3y - 5z = 4 x + 6y + 2z = -6
2) m + n + p + q = 7 2) Number of unknowns ________-2m + 4n – p + 3q = 14m – 2n + 4p + q = 4-m + 2n – 3p – 2q = 8
3) s + t – u = 5 3) Number of unknowns ________2s - 5t + 3u = 10 -s + 6t – 7u = 2
4) a + 2b + 3c + 4d + 5e = 6 4) Number of unknowns ________-a – 3b – 2c – 5d – 4e = 124a + 7b – 7c + 8d – e = -2-3a + 2b + 8c – 2e = 146a – 5b – 2c + d – 4e = 0
5) 2h – j + 4k – 2m = 23 5) Number of unknowns _________4h + 2j – k + 3m = -1h – 5j + 8k – 4m = 19-3h + j – 2k = -6
17
Name _______Answer Key_________________________ Date ________________
Solving Systems of Equations using Matrices
Directions: Solve the following systems of equations using the matrix method on your calculator.
1) 3x – y + 4z = -17 1) Number of unknowns ____3____4x + 3y - 5z = 4 x + 6y + 2z = -6x = -2.2731; y = .2098; z = -2.4927
2) m + n + p + q = 7 2) Number of unknowns ____4____-2m + 4n – p + 3q = 14m – 2n + 4p + q = 4-m + 2n – 3p – 2q = 8m = -3; n = 9; p = 11; q = -10
3) s + t – u = 5 3) Number of unknowns _____3___2s - 5t + 3u = 10 -s + 6t – 7u = 2s = 4 1/3; t = -1 2/3; u = -2 1/3
4) a + 2b + 3c + 4d + 5e = 6 4) Number of unknowns _____5___-a – 3b – 2c – 5d – 4e = 124a + 7b – 7c + 8d – e = -2-3a + 2b + 8c – 2e = 146a – 5b – 2c + d – 4e = 0a = 32.4846; b = 25.9596; c = 8.4963; d = -31.2455; e = 4.2179
5) 2h – j + 4k – 2m = 23 5) Number of unknowns _____4____4h + 2j – k + 3m = -1h – 5j + 8k – 4m = 19-3h + j – 2k = -6h = 2.4285; j = 6.5714; k = 2.6428; m = -7.0714
18
Day 6
Lesson: Given word problems, change into systems of equations and solve for the variables
Objectiveso Students will change word problems into systems of equations and solve for variableso Students will solve 2x2 and 3x3 systems using matrices on the calculatoro Students will recognize properties of systems of equations
Extra Materialso Class set of word problem worksheets
Opening Activity
Student Bellwork:The students have 5 minutes to complete the following problem. The students will then pass the paper to adifferent student and they will correct each others papers. The teacher will collect the papers after wecorrect them as a class.
Have students solve the following system using whichever method they choose:-3x + 4y = -2 Answers: x = -2.8-x + 2y = 6 y = 1.6
Main Activity
Given word problems, students will change into a system of equations and then solve for the variables. Wewill do some examples as a class, and some examples will be done as pairs. Students can use any method that wehave learned to solve the systems.
Closing Activity
Given this system, try to write a word problem that would make sense. You don’t need to solve the system,just write a word problem.
x = y + 107x – 3y = 25.60
Homework
Finish worksheet on word problems.
19
Worksheet on Word Problems
1. Five yards of fabric and three spools of thread cost $40.12. Two yards of the same fabric and ten spools of the same thread cost $23.88. Find the cost of a yard of fabric and the cost of a spool of thread.
Fabric _______________ Thread __________________
2. Half a watermelon and a half pound of cherries cost $3.09. A whole watermelon and two pounds of cherries cost $8.16.a. Write a system of equations that can be used to find the cost of each type of fruit.
b. Solve the system to find the cost of each type of fruit.
Watermelon ____________________ Cherries ___________________________
3. Two apples and six plums provide 300 calories. Three apples and five plums provide 350 calories.
How many calories are provided by five apples and eight plums?_________________________
20
4. At Wet Pets, a starter aquarium kit costs $15 plus $.60 per fish. At Gills and Frills, the same kit is $13 plus$.80 per fish.a. Give an equation for the cost c of f fish at each store.
Wet Pets ___________________ Gills and Frills ________________________
b. For what number of fish is the cost the same at the two stores? ______________________
5. For the Summer Rock Festival, there is one price for students, one for adults, and another for senior citizens. The Rueda family bought 3 student tickets and 2 adult tickets for $104. The Cosentinos bought 5 student tickets, 1 adult ticket, and 2 senior citizen tickets for $155. The Cragins bought 2 of each for $126.
a. Write a system of equations that can be used to find the cost of each ticket
b. Solve the system to find the cost of each ticket.
Students _______________ Adults _________________ Senior Citizens ___________________
6. A bicycle, three tricycles, and a unicycle cost $561. Seven bicycles and a tricycle cost $906. Five unicycles, two bicycles, and seven tricycles cost $1758.a. Set up a system of equations that can be used to find the cost of each item.
b. Solve the system to find the cost of each type of cycle.
Bicycle _________________ Tricycle ________________ Unicycle ________________
21
Worksheet on Word Problems Answer Key
1. Five yards of fabric and three spools of thread cost $40.12. Two yards of the same fabric and ten spools of the same thread cost $23.88. Find the cost of a yard of fabric and the cost of a spool of thread.
Fabric ____$7.49___________ Thread ____$.89______________
5f + 3t = 40.122f + 10t = 23.88
2. Half a watermelon and a half pound of cherries cost $3.09. A whole watermelon and two pounds of cherries cost $8.16.a. Write a system of equations that can be used to find the cost of each type of fruit.
.5w + .5c = 3.09w + 2c = 8.16
b. Solve the system to find the cost of each type of fruit.
Watermelon ______$4.20______________ Cherries ______$1.98 per lb._______________
3. Two apples and six plums provide 300 calories. Three apples and five plums provide 350 calories.
How many calories are provided by five apples and eight plums?_______575 calories_________
2a + 6p = 3003a + 5p = 350
a = 75 p = 25
5a + 8p = ?5(75) + 8(25) = 575
22
4. At Wet Pets, a starter aquarium kit costs $15 plus $.60 per fish. At Gills and Frills, the same kit is $13 plus$.80 per fish.a. Give an equation for the cost c of f fish at each store.
Wet Pets ___C = 15 + .60f________ Gills and Frills ___C = 13 + .80f_________
b. For what number of fish is the cost the same at the two stores? ______10 fish________
5. For the Summer Rock Festival, there is one price for students, one for adults, and another for senior citizens. The Rueda family bought 3 student tickets and 2 adult tickets for $104. The Cosentinos bought 5 student tickets, 1 adult ticket, and 2 senior citizen tickets for $155. The Cragins bought 2 of each for $126.
a. Write a system of equations that can be used to find the cost of each ticket3s + 2a = 1045s + a + 2c = 1552s + 2a + 2c = 126
b. Solve the system to find the cost of each ticket.
Students ___$18_________ Adults ____$25__________ Senior Citizens ____$20____________
6. A bicycle, three tricycles, and a unicycle cost $561. Seven bicycles and a tricycle cost $906. Five unicycles, two bicycles, and seven tricycles cost $1758.a. Set up a system of equations that can be used to find the cost of each item.
b + 3t + u = 5617b + t = 9062b + 7t + 5u = 1758
b. Solve the system to find the cost of each type of cycle.
Bicycle ____ $117________ Tricycle ____ $87________ Unicycle ___ $183_________
23
Day 7
Lesson: Given word problems, change into systems of equations and solve for the variables Quiz on Sections 5-2 to 5-4
Objectiveso Students will change word problems into systems of equations and solve for variableso Students will solve 2x2 and 3x3 systems using matrices on the calculatoro Students will recognize properties of systems of equations
Extra Materialso Class set of quizzes
Opening Activity
Quiz on Sections 5-2, 5-3, 5-4, 5-6 and including word problems
Main Activity
As a class, we will review how to change a word problem into a system of equations. We will go over indetail the previous night’s homework sheet answering any and all questions that may arise.
The students will first share their answers on the board:o One student will put the system of equations on the boardo Another student will solve the problem for the necessary information
This method will be repeated for each homework problem.
Closing Activity
Have students change this word problem into a system of 2 equations and hand in before exiting theclassroom:
At the zoo, Jay bought 3 slices of vegetable pizza and 1 small lemonade for $5.40.Terri paid $4.80 for 2 slices of vegetable pizza and 2 small lemonades.
Answer: 3P + 1L = 5.40 2P + 2L = 4.80
24
Name ________________________________________ Date ___________________Chapter Quiz Advanced Algebra
In questions 1 & 2, solve the system graphically or algebraically. Show all work.
1.ÓÌÏ
-=+
=-
235
56812
ts
ts1. _____________________
2.ÓÌÏ
+=
-=+
63
7419
xy
yx2. _____________________
3. Consider the system graphed below. How many solutions does the system have?
3. ______________________
In questions 4 & 5, refer the the following situation: At Federal Rent-A-Car, the cost of a one-day rental ofa midsize car is $45 plus $0.27 per mile driven. At Ready Rentals, the cost is $27 per day plus $0.36 per miledriven.
4. Let x = the number of miles driven and 4. ________________________ y = the cost of a one-day rental with x miles driven. Set up a system of two ________________________ equations to describe this situation.
5. a. For what number of miles driven will 5. a. ______________________ the cost of a one-day rental be the same at Federal Rent-A-Car and at Ready Rental.
b. What is the cost for this number of b. ______________________ miles driven?
*Extra Credit*
For what value of k does ÓÌÏ
=+
=+
829
122
yx
ykx have no solution?
Justify your answer.
__________________________________________________________________________________________
__________________________________________________________________________________________
25
Name ___________Answer Key_____________________ Date ___________________Chapter Quiz Advanced Algebra
In questions 1 & 2, solve the system graphically or algebraically. Show all work.
1.ÓÌÏ
-=+
=-
235
56812
ts
ts1. __s = 2; t = -4 ____________
2.ÓÌÏ
+=
-=+
63
7419
xy
yx2. ___x = -1; y = 3__________
3. Consider the system graphed below. How many solutions does the system have?
3. ________2 solutions____
In questions 4 & 5, refer the the following situation: At Federal Rent-A-Car, the cost of a one-day rental ofa midsize car is $45 plus $0.27 per mile driven. At Ready Rentals, the cost is $27 per day plus $0.36 per miledriven.
4. Let x = the number of miles driven and 4. ____y=45+.27x___________ y = the cost of a one-day rental with x miles driven. Set up a system of two ____y=27+.36x___________ equations to describe this situation.
5. a. For what number of miles driven will 5. a. ___200 miles___________ the cost of a one-day rental be the same at Federal Rent-A-Car and at Ready Rental.
b. What is the cost for this number of b. ___$99.00_____________ miles driven?
*Extra Credit*
For what value of k does ÓÌÏ
=+
=+
829
122
yx
ykx have no solution?
Justify your answer.
____When k = 9, there is no solution because when you perform linear combinations on the system you get that0=4 and this will never be true, therefore the lines are parallel, and no solution exists. _____________________
