Download Objective: SWBAT solve systems of equations using the

Algebra 1: Unit 5: Systems of Equations
Lesson Title
Content Objectives
5.1: Linear
Equation
Review
* This is an
optional lesson.
SWBAT graph, solve and write
linear equations
5.2: Solving
Systems of
Equations with
Graphing
SWBAT solve systems of equations
using the graphing method.
5.3: Solving
Systems of
Equations With
Substitution
5.4: Solving
Systems of
Equations with
Elimination
5.5: Solving
Systems of
Equations with
Matrices
Language Objectives
Learning
Standard
DP
Time
Days
SWBAT write a paragraph
differentiating between input and
output variables in linear equations
and explain how the slope and yintercept influence the aspects of a
line.
SWBAT explain why a system of
linear equations can have one
solution, no solution, or infinite
many solutions and describe what
the graph of each situation looks
like.
A-CED.1
A-CED.2
A-REI.3
M01 2
.
M03
.
A-CED.3
A-REI.6
M01 2
.
M03
.
E09.
SWBAT solve systems of equations
using the substitution method.
SWBAT explain the steps involved in
solving a system of linear equations
using the substitution method.
A-REI.6
SWBAT solve systems of equations
using the elimination method.
SWBAT explain the steps involved in
solving a system of linear equations
using the elimination method.
A-REI.6
SWBAT solve systems of equations
using matrices.
SWBAT explain what a matrix is and
how it can be used to solve a system
of linear equations or to check a
solution.
A-REI.6
A-REI.8
A-REI.9
M03 2
M05
.
E09.
M01 2
.
M03
.
E09.
M05 3
.
E09.
5.6: Evaluating SWBAT apply their knowledge of
Methods of
solving systems of equations and
Solving Systems justify when to use which method
of Equations
SWBAT describe the best scenario in A-CED.3
which to use one of the four methods A-REI.6
for solving system of linear
equations.
M01 1
.
M03
.
E09.
T04.
Lesson 5.1: Linear Equation Review
Content Objective: SWBAT graph, solve and write linear equations
Diploma Plus Competencies:M01. Problem Solving, M03. Quantitative Reasoning,
*Note: The focus of this lesson is to recall and apply computational Algebra 1 S1 concepts.
In order to understand a relationship between two variables we can make a table of values. (Remember: a
variable is a value that changes.)
Consider the equation y = 4x + 1. This equation has two variables: x and y. The two variables are related by the
equation. We can choose values of x and find out how the y changes. You can also think of the x and an input.
Choose and input number for x and the y is the output. We will choose 5 values for x (-2, -1, 0, 1, 2) and plug them
in for x into the equation.
.
X
-2
-1
0
1
2
Y
4(-2) + 1 = -7
4(-1) + 1 = -3
4(0) + 1 = 1
4(1) + 1 = 5
4(2) + 1 = 9
Use this table to create a graph for the equation y = 4x + 1.
From the graph you should notice some correlation between the equation and the way the graphs look. First, these
equations create a striaght line. Equations of the form y = mx + b will always make straight lines in a graph, which
is why they are called Linear Equations (note: the m and the b in the equation are numbers). Second, the numbers
m and b in the equation describe what the graph will look like. The number multipling x, m, is called the slope; and
the number b is called the y-intercept.
Slope is a measure of how much the graph is increasing or decreasing. Slope is most easily understood as a
fraction.
Slope 
Rise change in y

Run change in x
Notice how in the equation y = 4x + 1 when the x value changes +1 then the y value changes +4.
Slope 
Rise change in y 4

 .
Run change in x 1
Practice:
X
Make a table and a graph for each of the following equations.
X
Y
Y
Practice Problems: Holt McDougal Algebra 1 Textbook: pages 246-247 problems #26-34, 43-48.
For additional support please see the resources section of Summit’s Math Site.
The second skill you will need to use during this unit is solving equations. Here is a quick review:
Example:
You work for a contractor and make $900 per week. You started the year with $1100 in your bank
account. How many weeks until you have a total of $10,000 in your bank account?
First you need to write the linear equation for this situation (y = mx +b). Let y stand for the amount
of money in your account, and x is the number of weeks since the beginning of the year.
Y = 900x + 1100
We want $10,000 in out account, we can plug in this value for y
10000 = 900x + 1100
Now we can solve for the remaining variable x,
-1100
- 1100
Subtract 1100 from both sides
9900 = 900x
Divide both sides by 900
900 900
11 = x
It will take 11 weeks until you have $10,000 in your account.
Practice:
Solve the following equations.
Practice Problems: In the Holt McDougal Algebra 1 Textbook solve the following problems:
#49 and #50 on page 142; #8 and #15 on page 149; #48 and #49 on page 168; #15 on page 149
Higher Level Thinking Questions:
Language Objective: SWBAT write a paragraph differentiating between input and output variables in linear equations
and explain how the slope and y-intercept influence a line.
Write a paragraph answering the following questions:
1. What is a variable and explain the difference between an input and output variable.
2. What is a linear equation and explain how the output variable in a linear equation is related to the input
variable.
3. Explain how the slope and y-intercept of a linear equation affect what the line looks like.
Lesson 5.2: Solving Systems of Equations with Graphing
Objective: SWBAT solve systems of equations using the graphing method.
Diploma Plus Competencies: M01. Problem Solving, M03. Quantitative Reasoning, E09. Language Choice.
Now we can use graphs to solve systems of linear equations.
A system of equations is a collection of two or more equations with a same set of unknowns. In solving a
system of equations, we try to find where the two equations will intersect or cross.
When we solve systems of equations by graphing, we will:
1) Graph both lines on a coordinate plane (Using slope intercept form or standard form)
2) Label where they intersect.
Example
Let’s try and solve systems of equations in word problems.
Here is an example from Discovering Algebra: Second Edition:
Practice:
Use the graphing method to solve the following systems of equations. When finished, identify the solution, or
the point of intersection, as an ordered pair. If you lines crossed when x=3 and y= -2 then your ordered pair
would look like (3, -2).
For additional support please see the resources section of Summit’s Math Site.
Solution:
Solution:
Solution:
Solution:
Practice:
1. Two friends start rival hair salons. It costs Cutz $1000 to set up the salon and they make $6.00 a haircut.
a) Write a linear equation to describe the profits for Cutz. _______________________________________________
b) The other friend opens up another salon named Clipz. Their profit equation is y=8.50x-1500. Explain the
meaning of each variable and number in the equation. _______________________________________________
c) Graph both profit equations and find the solution.If you graphed the system by hand staple your graph to
the back of this exploration. Solution: ____________________________________________________
d) Explain the real world meaning of the solution of this problem.
2. Adapted from Discovering Algebra: Second Edition:
The costs for two families to attend the basketball game are given by: 2x + 3y = 13.50 and 3x + 2y = 16.50. Where x
is the cost of an adult ticket and y is the cost of a student ticket in dollars.
a) What is the real world meaning of the first equation? _____________________________________________________________
b) Solve this system of equations using graphing. If you graphed the system by hand staple your graph to the
back of this exploration. Solution: ______________________________________________
c) Explain the real world meaning of the solution of this problem.
4. Adapted from Discovering Algebra: Second Edition:
The manager of a movie theatre wants to know the number of adults and children who go to the movies. The
theater charges $8.00 for adults and $4.00 for children. At a showing of Iron Man 3, 200 tickets were sold and the
theater made $1304.00.
a) Let the variable A represent the number of adult tickets and the variable C represent the number of child
tickets. Write an equation to for the total number of tickets sold. _______________________________________________
b) Write an equation showing the total cost of the tickets. __________________________________________________________
c) Graph the system of equations and find the solution. If you graphed the system by hand staple your graph
to the back of this exploration. Solution: ___________________________________________________________________________
d) Explain the real world meaning of the solution of this problem.
Higher Level Thinking Questions:
Language Objective: SWBAT explain why a system of linear equations can have one solution, no solution, or
infinitely many solutions and describe what the graph of each situation looks like.
Write a paragraph answering the following questions:
1. Explain why a system of linear equations can have either one solution, no solution, or infinitely many
solutions.
2. Describe what the equations and the graphs look like for each situation: one solution, no solution, infinitely
many solutions.
Lesson 5.3: Solving Systems of Equations with Substitution:
Objective: SWBAT solve systems of equations using substitution.
Diploma Plus Competencies: M03. Quantitative Reasoning, M05. Communication, E09.Language Choice.
The method of solving by substitution works by plugging in information about one variable back into the other
equation, substituting for the chosen variable and solving for the other. Then you back-solve for the first variable.
Here is how it works.
STEPS:
1) Make sure one of the equations is solved for 1 variable
2) Substitute that variable into your other equation.
3) Solve for the second variable.
4) Plug in what you just solved for in either equation.
5) Solve for remaining variable.
Solve the following system by substitution.
2x – 3y = –2
y = -4x + 24
In this problem one of the equations is solved for y so I can use the SUBSTITUTION METHOD to solve and find the
place where both of these lines intersect.
From the second equation I know that y = -4x + 24. This means I can substitute -4x + 24 in for y in the first
equation. And then solve for x.
2x – 3(–4x + 24) = –2
2x + 12x – 72 = –2
14x = 70
x=5
Distributive Property
Combine like terms and add 72 to both sides
Divide both sides by 14
Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
Now I can plug this x-value back into either equation, and solve for y.
y = –4(5) + 24
y = -20 + 24
y=4
Then the solution is (x, y) = (5, 4).
In Discovering Algebra read Example A on page 281 and 283 – 284 for additional support.
For additional support please see the resources section of Summit’s Math Site.
Practice:
1. Adapted from Discovering Algebra: Second Edition:
This system of equations models the profits of two shoe stores.
P = 25N - 12000
P = 16N – 5000
The variable P represents the profit in dollars, and N represents the numbers of pairs of shoes sold.
a) Use the substitution method to find an exact solution.
Solution: __________________________________________________
b) Explain the real world meaning of the solution of this problem.
Higher Level Thinking Questions:
Language Objective: SWBAT explain the steps involved in solving a system of linear equations using the
substitution method.
Write and explain the steps necessary to solving the following system of equations:
y = 5x – 3
2x + 6 = -10
Lesson 5.4: Solving Systems of Equations with Elimination
Objective: SWBAT solve systems of equations using the elimination method.
Diploma Plus Competencies: M01. Problem Solving, M03. Quantitative Reasoning, E09. Language Choice.
Example
Adapted from Discovering Algebra: Second Edition
The school’s photographer took pictures at the prom. She charged $3.25 wallet and $10.50 for portrait.
Crystal and Diego bought a total of 10 pictures for $61.50.
a. Write the system of equations for this scenario.
b. Solve the system of equations with elimination and explain the meaning of the solution.
Solution:
a.
b.
Let w represent the number of wallet pictures and p represent the number of portrait pictures.
Then the system is:
w + p = 10
3.25w + 10.50p = 61.50
The first equation describes the total number of pictures. The second equation describes the cost of
the pictures bought.
Notice that we can make the coefficient for w in the first equation the same as the coefficient of w in
the second equation with multiplication
First Equation:
w + p = 10
multiply by 3.25
3.25w + 3.25p = 32.50
Now that the coefficients of w for each are the same we can use subtraction to eliminate the w
variable.
First Equation:
3.25w + 3.25p = 32.50
Second Equation:
- 3.25w + 10.50p = 61.50
subtract the two equations
0 w + -7.25p = -29
0w = 0
-7.25p = -29
solve for p
p =4
divide both sides by -7.25
Now that we know p = 4 we can plug this value into one of our first two equations and solve for w.
First Equation:
w + 4 = 10
w=6
remember p = 4
solve for w
Solution w = 6 , p =4 (6, 4). This means that Crystal and Diego bought 6 wallet pictures and 4
portrait pictures.
For additional support please see the resources section of Summit’s Math Site.
Practice:
Higher Level Thinking Questions:
Language Objective: SWBAT explain the steps involved in solving a system of linear equations using the
elimination method.
Write and explain the steps necessary to solving the following system of equations:
6x – y = 4
2x + 2y = -10
Lesson 5.5: Solving systems of equations with matrices:
Objective: SWBAT solve systems of equations using matrices.
Diploma Plus Competencies: M01. Problem Solving, M03. Quantitative Reasoning, E09. Language Choice.
To learn how to solve systems of equations with matrices please watch the Matrices Video on Summit’s Math Site
and read pages 296-299 in Discovering Algebra: Second Edition.
Practice:
From Discovering Algebra: Second Edition
Higher Level Thinking Questions:
Language Objective: SWBAT explain what a matrix is and how it can be used to solve a system of linear equations
or to check a solution.
Write and explain the steps to solving a matrix using a TI-84 calculator.
Lesson 5.6: Evaluating Methods of Solving Systems of Linear Equations
Objective: SWBAT apply their knowledge of solving systems of equations and justify when to use which method
Diploma Plus Competencies: M01. Problem Solving, M03. Quantitative Reasoning, E09. Language Choice, T04.
Critical Thinking, Problem-Solving & Decision-Making
To help you decide which method (graphing, substitution, or elimination) to use solving a system of linear
equations, consider the following.
 THE GRAPHING METHOD


o
Useful for approximating a solution, checking a solution and for providing a visual model for the
problem.
o
Best used when both equations are in slope-intercept form.
o
Remember that parallel lines have no solutions and two different forms of the same equation will
represent the same line and have infinite solutions.
THE SUBSTITUTION METHOD
o
Good to find an exact solution
o
Best used when at least one of the equations is solved for a variable.
THE ELIMINATION METHOD
o
Good to find an exact solution
o
Best used when both equations are in standard form. (ax + by = c)
Higher Level Thinking Questions:
Language Objective: SWBAT describe the best scenario in which to use one of the four methods for solving system
of linear equations.
Complete the following table with complete sentences
Graphing
When would
you use it?
(What form
should the
equations be in
to justify using
the given
method?)
Substitution
Elimination
Matrix