UNIT PLAN:
SOLVING SYSTEMS OF EQUATIONS
Course – Algebra 1
Grade Level – 9th
Time Span – 6 Days, 80 minute periods
Tools – Algebra tiles, TI Graphing Calculators, Internet & class set of
computers, projector for teacher computer, Alg Chapter 5 APP for TI
graphing calculators, Green Globs
BY:
AMANDA STEPHENS
FALL 2006
AUDIENCE: Algebra 1
TIME FRAME: 10 40-minute period.
NYS PERFORMANCE STANDARDS:
AA7 Analyze and solve verbal problems whose solution requires solving systems of
linear equations in two variables.
AA10 Solve systems of two linear equations in two variables algebraically.
AG7 Graph and solve systems of linear equations and inequalities with rational
coefficients in two variables.
ACM10 Solve systems of linear equations using concrete models, graphs, tables,
algebraic methods and check the reasonableness of solutions.
NCTM STANDARDS:
• Understand vectors and matrices as systems that have some of the properties of
the real-number system
• Develop an understanding of properties of, and representations for, the addition
and multiplication of vectors and matrices
• Develop fluency in operations with real numbers, vectors, and matrices, using
mental computation or paper and pencil calculations for simple cases and
technology for more complicated cases.
• Judge the reasonableness of numerical computations and their results.
OBJECTIVES:
By the end of this unit, students will be able to:
1. Solve systems of equations using various methods including substitution,
elimination using addition and subtraction, and elimination using multiplication,
2. Use a graphing calculator to solve systems of equations by graphing and matrices.
RESOURCES:
Holliday, Cuevas, Moore-Harris, Carter, Marks, Casey, Day, Hayek, Algebra 1; New
York State Edition, McGraw Hill, Glencoe, New York, 2006.
http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=syslin, Math A Regents
Prep, Donna Roberts.
http://mathbits.com/MathBits/TISection/Precalculus/matrices.htm, Matrices Using the
Graphing Calculator, Frederick & Donna Roberts.
http://www.analyzemath.com/Tutorial-System-Equations/Tutorial-SystemEquations.html. Solve Systems of Equations-Tutorial. Dr Abdelkader Dendane, Ph.D.
MATERIALS:
Graphing calculators
Algebra tiles
Graph paper (incl)
Worksheets (incl)
Rulers
Markers
TI APP – Algebra Ch. 5
Glencoe, Algebra, NY Edition, Ch. 7
2
Quiz (incl)
Smartview & Overhead
Green Globs software
Internet and Projector
OVERVIEW
DAY 1
Introduce idea of what is a system of equations
Solve systems of equations by graphing
Use graphing calculators & graph paper
Key Vocab: system of equations, consistent, inconsistent, dependent,
independent, intersection
DAY 2
Solve systems by substitution
Use Algebra Tiles
Key Vocab: substitution
DAY 3
Solve systems by elimination using addition or subtraction
Key Vocab: Elimination
DAY 4
Solve systems by elimination using multiplication
DAY 5
Solve systems using matrices
Use Graphing calculators
DAY 6
Complete review/ take up review sheet
Alg Ch 5 App from TI may be used to help review
Quiz
*NOTE: Days can be rearranged; only elimination by addition should be taught before
elimination by multiplication.
Use Algebra Ch 5 APP from TI at any point suitable.
This app reviews consistent and independent, consistent and dependent, and inconsistent.
It also explains the elimination method and has a couple of games to help students master
the subject.
ANTICIPATORY SET:
Marcus went to the movies and bought 2 tubs of popcorn and 1 soda for $4.75. Jackie
went to the movies and bought 1 tub of popcorn and 2 sodas for $4.25. What is the cost
of 1 soda and 1 popcorn?
DAY 1 – GRAPHING
Objectives
Solving systems by graphing.
ATTACHED: Graph paper, graphing activity, graphic organizer to help with
terminology.
Students should be familiar with graphing a line on the graphing calculator. If they are
not, they need to know how to input a linear equation and how to read the table to plot
the ordered pairs.
3
Key Terms
In general, a solution of a system in two variables is an ordered pair that makes
BOTH equations true.
In other words, it is where the two graphs intersect, what they have in common. So if an
ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the
system.
A consistent system is a system that has at least one solution.
An inconsistent system is a system that has no solution; parallel lines.
The equations of a system are dependent if ALL the solutions of one equation are also
solutions of the other equation. In other words, they end up being the same line.
The equations of a system are independent if they do not share ALL solutions. They
can have one point in common, just not all of them.
Use green globs to show examples of the different types of systems and work on
vocabulary. Graph the following systems then discuss the solutions and the terminology in
the chart below:
(*note – these are not in y=mx+b format as Green Globs does not require this format –
decide whether this is a good time to work on solving for y or to work on the vocabulary;
students need to be accustomed to seeing the equations in different forms and know that
they can still solve them!)
1) x+2y=3; 3x-y=-5, 1 solution, consistent and dependent.
2) y=2x-3; 4x=2y+6, Infinite solutions, consistent and dependent.
3) 3x-y=-2; 3x-y=0, no solution, inconsistent; lines are parallel.
Have students sketch the three different examples in their notebooks and label each with the
appropriate vocabulary. The note should resemble the chart below.
INTERSECTING
LINES
SAME LINE
PARALLEL
LINES
One Solution
Infinite Solutions
No Solution
Consistent
Independent
Consistent
Dependent
Inconsistent
GRAPH
# OF
SOLUTIONS
TERMINOLOGY
Use the following websites as a reference or to teach students how to use the graphing
calculator.
http://www.mathbits.com/MathBits/TISection/Algebra1/systems.htm - step by step
instructions with visuals on solving systems using the graphing calculator. This is good
because it shows the calculator screens.
http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=syslin - step by step
instructions with problems for students to work out using the graphing calculator. This
site is nice because it leads right into hands on problems for students.
4
If the above websites are not used, the following should be completed.
Using the graphing calculator, the equations must be in y=mx+b form. In Y= type in
both equations. Hit GRAPH to show the lines. To find the point of intersection, the
point needs to be on the viewing window. This may require adjusting the window or
hitting zoom, fit. Then, find the intersection by 2nd Trace (Calc), choose 5 Intersection.
Follow the prompts on the screen, the calculator should move from curve to curve as
enter is hit but it may not if it is close to the side of the window, it may need manually
moved from curve to curve by using the right and left arrows. After choosing curve 1,
curve 2, and guess, the point of intersection will be shown.
The table (2nd Graph) can also be used if the solution(s) is a whole number. The table can
reset (2nd window, change in TBL) the change in values to view decimal solutions,
however, this gets tricky for a level 1 algebra course. The solution is where the x value
has two of the same y values. For example if x is 2 and y1 is 5 and y2 is 5, the solution is
(2, 5). Students should try to estimate the answer before going to the table as it will make
the solution easier to locate.
Use the smartview to show the students the steps on the overhead. Do examples that the
window will need to be adjusted. If students have a difficult time adjusting the window,
they can use zoom fit. Zoom standard is also a good key for them to remember as it
brings back the standard viewing window.
Ex. 1
Y=-6; 4x+y=2
- solve both equations for y
- graph & find intersection (solution)
- describe the solution
Ans – y=-6, y=-4x+2; (2, -6); one solution, consistent and independent.
Ex. 2
X+y=24; 3x+y=44
- solve both equations for y
- graph & find intersection (adjust window; xmax 15, ymax 50 or zoom fit)
- describe the solution
Ans – y=-x+24, y=-3x+44; (10, 14); one solution, consistent and independent
Pass out graph paper (attached below) and have students graph, find the solutions, and
describe the solutions to the following systems.
1. 2x-y=-3; 8x-4y=-12; Ans - infinite solutions, consistent, dependent
2. x-2y=4; x-2y=-2; Ans -no solution, inconsistent
3. x+y=3; 12x+4y=20; Ans – (1,2), one solution, consistent, independent
5
DAY 2 – SUBSTITUTION
Objectives
Solve systems of equations using substitution.
Introduce solving systems by substitution. Explain that substitution means replacing one
variable so that there is only one variable to solve for instead of two. Give students an
example of substitution in real life. If I have 2 coke and 1 pepsi, then replace the pepsi
with a coke, now I have only coke. The main thing the students need to understand is
that by substituting, they end up with only 1 variable.
Model this method using algebra tiles. Use examples that solve for x first, that is, y is
replaced.
Ex. 1 3x+y=8, y=x-4; since y is x-4, replace y in the first equation with x-4. Using
parenthesis whenever anything is replaced is helpful when students need to distribute so
they have it set up correctly. In this problem, 3x+(x-4)=8. (Solution: (3,-1))
3X
{
XX
XX
X
X-4
{
X
-1
-1
-1
-1
X
=
} 8
Solve like a regular equation. Students can have the equation mats and also a worksheet
where they solve the equation step by step. Use the value for x to solve for y by
substituting x into either of the original equations.
Ex. 2 4x+3y=10 and y=x+1 (Solution: (1,2))
Use the algebra tiles to solve this problem. Write each step out on paper.
Sometimes, one equation is not solved in terms of the other variable. In these problems,
first you must solve one equation so that it equals a variable. After one equation is
solved, use the substitution method.
Ex. 3 4x+y=12, -2x-3y=14 (Solution: (5, -8))
In this problem, since the y in equation 1 is equal to one, it is easier to solve this equation
in terms of y; y=-4x+12. Substitute this new value for y into equation 2 where y is used.
DON’T forget the parenthesis!! -2x-3(-4x+12)=14. Solve for x. Use this value of x to
solve for y.
The website may be used to help teach this method or as a review after it is taught.
http://regentsprep.org/Regents/math/syslin/AlgSys.htm - steps to solving equations using
substitution
Solve the following problems in student notebooks:
1. x=4y, 4x-y=75 (20, 5)
2. y=3x-8, y=4-x (3, 1)
3. x+3y=12, x-y=8 (9, 1)
6
4. x+14y=84, 2x-7y=-7 (14, 5)
5. 0.5x+4y=-1, x+2.5y=3.5 (6, -1)
6. At the end of the 2000 baseball season, the NY Yankees and the Cincinnati Reds
had won a total of 31 World Series. The Yankees had won 5.2 times as many
World Series as the Reds. How many World Series did each team win?
(Y+R=31, Y=5.2R; y-26, r-5).
Take up these problems in class if time permits. Homework sheet is attached.
DAY 3 – ELIMINATION BY ADDITION/SUBTRACTION
Objectives
Solve systems of equations using elimination by addition and subtraction.
Introduce solving by elimination using addition or subtraction. Elimination means
getting rid of something, in this case, remove a variable. Again, the idea is to have only
one variable to solve for.
http://regentsprep.org/Regents/math/syslin/AlgSysAdd.htm - steps to solving systems of
equations using elimination.
Given:
x+y=62
+ x -y=12
2x =74
2
2
x=37
Since the y values have a coefficients of positive &
negative one, add them together to eliminate y.
Using x=37, substitute this new value for x into either of the original equations.
37 + y =62
-37
-37
y =25
Therefore, the solution is x=37 and y=25.
Given:
a+b=15
- a+2b=3
-1b=12
-1 -1
b=-12
Since the a values have coefficients of positive one,
subtract to eliminate a.
Using b=-12, substitute this new value for b into either of the original equations.
A+(-12)=15
+12 +12
a = 27
7
Students try:
1) a-w=120; a+w=150
Ans – a=135, w=15
2) 4x+13y=40; 4x+3y=-40 Ans – x=-16, y=8
3) A resort hotel offers two weekend specials; Plan A is 3 nights with 6 meals for $264,
Plan B is 3 nights with 2 meals for $218. What is the cost of one night’s lodging and
what is the average cost per meal?
Solution:
3N+6M=264
- 3N+2M=218
4M=46
4 4
M=11.5 The cost of a meal is $11.50.
Substitute m=11.5 into either original equation;
3N+6(11.5)=264
3N+69
=264
-69
-69
3N
=195
3
3
N =65 The cost for one night is $65.00.
Complete worksheet below in class or for homework.
DAY 4 – ELIMINATION BY MULTIPLICATION
Objectives
Solve systems of equations using elimination and multiplication.
This method is similar to elimination by addition and subtraction, however one more step
is added. If the coefficients of one of the variables are not the same, find a common
multiple for one set and multiply each term by that number. Do the same for the second
equation. In the end, one set of variables needs to have the same coefficient, either
positive or negative. Students may want to multiply by a negative so they can add rather
than subtract the two systems.
http://www.analyzemath.com/Tutorial-System-Equations/Tutorial-SystemEquations.html - a tutorial including step by step instructions on solving systems of
equations by multiplication and elimination.
Assignment:
Glencoe – Algebra 1 chapter 7-4, p.390, #1-12.
DAY 5 – MATRICES
Objectives
Solve systems of equations using matrices.
The following website may be used to show students how a set of matrices can be used to
solve systems of equations.
8
http://mathbits.com/MathBits/TISection/Precalculus/matrices.htm - steps to inputting
matrices onto a calculator and how to solve the system using matrices.
Students will want a copy of the steps to inputting a matrix system into their calculator
and how to solve it.
Steps to Using a Matrix to Solve Systems of Equations
*Make sure the variables and numbers are lined up vertically underneath one another.
2m+3n=9
5m+4n=5
Matrix A is 2X2
2 3
5 4
Matrix B is 2X1
9
5
On the calculator, type 2nd x-1, move over to edit, choose 1 or A. Input the data, ie 2 enter
2 enter 2 enter 3 enter 5 enter 4 enter.
Input matrix 2. Type 2nd x-1, move over to edit, choose 2 or B. Input the data; 2 enter 1
enter, 9 enter, 5 enter.
Go back to home screen, 2nd Mode (Quit).
Type in the inverse of A multiplied by B. 2nd x-1 1 or A, x-1 * 2nd x-1 2 or B.
The answer is in matrix form. The first variable is on top, m=-3 and the second variable
is on the bottom, n=5.
Try the following problems from the worksheet “Solving Systems using Matrices” using
the steps above.
As a final review, students need to be able to solve equations given any format. A work
sheet is attached (Systems of Equations – look for the tiger) that can be used in class or as
homework. (No word problems)
http://www.mathbits.com/MathBits/TeacherResources/Algebra/SolvingSystems.pdf review of all formats of systems of equations. (No word problems)
ASSESSMENT – See attached quiz
9
HMWK – Solving Systems by Substitution
Please solve the following systems of equations for x & y using the substitution method.
Check your work.
1. y=3x
x+2y=-21
2. x=4y
2x+3y=44
3. x+5y=-3
3x-2y=8
4. c-5d=2
2c+d=4
5. 5r-s=5
-4r+5s=17
10
SYSTEM OF
EQUATIONS
Solution Exists
No Solution
CONSISTENT
One
Solution
INDEPENDENT
INCONSISTENT
Many
Solutions
DEPENDENT
11
12
SYSTEM OF
EQUATIONS
Solution Exists
One
Solution
No Solution
Many
Solutions
13
GRAPHING SYSTEMS OF EQUATIONS – INVESTIGATION WORKSHEET
Veronica is considering two job offers in car sales. The first job pays $400 per week plus
15% sales commission. The second job pays $350 per week plus 20% sales commission.
What amount would Veronica have to sell for her weekly salary to be the same at either
job?
Problems:
1. Write two equations, one for each job, representing the amount of weekly sales, x,
and total weekly salary, y.
2. Type these two equations into Y1 and Y2 in the graphing calculator. Graph the
functions. What is on the screen? How does the viewing window need to be
adjusted?
3. Find the point of intersection of the two lines. What is the ordered pair?
4. At what amount of sales will the two jobs have the same salary? What will that
salary be?
Solution – 2. xmax – 1200, ymax – 700 3. (1000, 550) 4. Sales $1000, Salary $550
14
Solving Systems of Equations
Elimination by Addition & Subtraction
Solve the following systems of equations using elimination by addition or subtraction.
1. 2x+8y =2
-2x-4y=6
2. a+b=11
a-b=4
3. 2x-3y=5
5x-3y=11
4. 2m+n=-5
2m+3n=7
5. Suppose two eggs with bacon cost $2.70. One egg with bacon costs $1.80. At
these rates, what will bacon alone cost?
6. Five gallons of regular unleaded gas and eight gallons of premium gas cost
$17.15. Five gallons of regular unleaded and two gallons of premium gas cost
$8.75. Find the cost per gallon of each kind of gasoline.
15
Answers – Solving Systems of Equations using Elimination by addition &
subtraction.
1.
2.
3.
4.
5.
6.
x=-7 y=2
a=7.5 b=3.5
x=2
y=-1/3
m=-5.5 n=6
eggs=.90 bacon=.90
regular = 1.19 premium=1.40
16
Solving Systems of Equations
Matrices
Use the graphing calculator to solve the following systems of equations using matrices.
Show your work and the key sequence of the calculator steps for full credit!
1. 2x+3y=15
5x-2y=-29
2. 5x=4y+15
6y=3x-9
3. The sum of two numbers is 45. Three times the first number plus seven times the
second is 115. Find the two numbers.
4. Tickets for a school play sell for $3 for a student and $5 for an adult. One night 595
people bought tickets. The school took in $1951. How many adult tickets and how many
student tickets were sold?
5. Mary has 30 nickels and dimes totaling $2.40. How many of each does she have?
17
ANSWERS – Systems of Equations using Matrices
1. x=-3, y=7
2. 5x-4y=15, -3x+6y=-9; x=3, y=0
3. x+y=45, 3x+7y=115; x=50, y=-5
4. s+a=595, 3s+5a=1951; s=512, a=83
5. n+d=30, 5n+10d=240; n=12, d=18
18
Systems of Equations
Quiz
1. Graph the system of equations. Then determine whether the system has no solution,
one solution, or infinitely many solutions. If the system has one solution, name it.
Use substitution to solve the system of equations.
2.
Use elimination to solve the system of equations.
3.
4. At a local electronics store, CDs were on sale. Some were priced at $14.00 and some at
$12.00. Sabrina bought 9 CDs and spent a total of $114.00. How many $12.00 CDs did
she purchase?
19
Systems of Equations
Answer Section
SHORT ANSWER
1. ANS:
one solution; (–3, –5)
Graph each line. The point where the two lines intersect is the solution. Check the
solution by replacing x and y in the original equations with the values in the ordered pair.
DIF: Average
OBJ: 7-1.2 Solve systems of equations by graphing.
STO: A.A.7, A.G.7
TOP: Solve systems of equations by
graphing
KEY: System of Equations, Graphing
MSC: 1998 Lesson 8-1
NOT: /A/ Remember that the x-coordinate comes first in an ordered pair. /B/ Did you
graph the second line correctly? /C/ Graph both lines./D/ Correct!
2. ANS:
(33, 36)
Substitute the right side of the top equation for y in the bottom equation. Solve for x.
Substitute the value obtained for x into the top equation and solve for y.
DIF: Average
OBJ: 7-2.1 Solve systems of equations by using substitution.
STO: A.A.7, A.A.10, A.G.7
TOP: Solve systems of equations by using substitution
KEY: System of Equations, Substitution MSC: 1998 Lesson 8-2
NOT: /A/ Remember to list the x-coordinate first and then the y-coordinate./B/ Use
parentheses when you sustitute for y and then remember to distribute the negative sign
through the parentheses. /C/ Terms with different variables cannot be combined. /D/
Correct!
3. ANS:
(–1, –1)
20
Eliminate one variable by adding the two equations. Solve for x and then substitute that
value into one of the equations to find the value of y.
DIF: Average
OBJ: 7-3.1 Solve systems of equations by using elimination with
addition.
STO: A.A.7, A.A.10, A.G.7
TOP: Solve systems of equations by using elimination with addition
KEY: System of Equations, Elimination, Addition
MSC: 1998 Lesson 83
NOT: /A/ Correct! /B/ Double-check your positive and negative signs. /C/ Add the
equations together./D/ Add the equations together.
4. ANS:
6
Solve the first equation for one of the variables and substitute into the second equation.
Solve. Substitute that value into the first equation to find the second value.
DIF: Average
OBJ: 7-2.2 Solve real-world problems involving systems of
equations.
STO: A.A.7, A.A.10, A.G.7
TOP: Solve real-world problems involving systems of equations
KEY: System of Equations, Real-World Problems
MSC: 1998 Lesson 82
NOT: /A/ This is the total number of CDs purchased. /B/ Correct! /C/ Does your answer
satisfy the equations? /D/ This is the number of $14.00 CDs purchased.
21
22
REPRODUCED FROM:
http://www.mathbits.com/MathBits/TeacherResources/Algebra/SolvingSystems.pdf
23