Download Unit 6 – Systems 6–1 Graphing Systems of Equations 6–2

Unit 6 – Systems
6–1 Graphing Systems of Equations
6–2 Substitution
6–3 Adding and Subtracting
6–4 Multiplication
6–5 Graphing Systems of Inequalities
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Section 6-1: Graphing Systems of Equations (Day 1)
Review Question
What makes an equation linear? Exponents on variables are 1
What is a solution to a linear equation? Point; y = 3x +1 (2, 7)
Discussion
What do you think a system of equations is? 2 or more equations
What is a solution to a system of equations? Point that works in all equations
y = 2x – 3
y = -3x + 7
Notice that (2, 1) works in both equations.
What would that look like? Two lines
(2, 1) is where the two lines would intersect.
How many ways can two lines intersect?
# of Intersections
# of Solutions
How?
1
1
Different m’s
0
0
Infinite
Infinite
Same m’s;
Different intercepts
Same m’s;
Different intercepts
SWBAT find the solution to a system of equations by graphing
Example 1: Graph each line to find the solution.
y = 2x – 3
y = -3x + 7
(2, 1)
How do you know that your answer is correct? (2, 1) “works” in both equations
Example 2: Graph each line to find the solution.
y = 4x – 3
y = -3x + 7
(1.5, 3)
How do you know that your answer is incorrect? (2, 2) doesn’t “work” in either equation
Hmmmm?!?
What issue do you see with graphing to find the solution? It is not exact.
Example 3: Graph each line to find the solution.
y = 2x – 3
y = 2x + 3
No Solution
What does the answer of No Solution mean? No numbers will work in both equations.
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Example 4: Graph each line to find the solution.
y = 2x + 1
2y – 4x = 2
Infinite Solutions
What does the answer of infinite solutions mean? There are an infinite amount of answers
that will work in both equations.
You Try!
Graph each line to estimate the solution.
1. y = 2x + 3
y = -3x + 1
(.5, .5)
2. y = 4x – 1
y – 4x = 2
No Solution
3. y = 4x + 1
3y = 12x + 3
Infinite Solutions
4. y + 3x = 1
(1, -3)
2
y  x 3
3
________
5. 3y – 8 = 4x
x=2
(2, 5)
6. y = 5
y = -2
No Solution
What did we learn today?
Section 6-1 Homework (Day 1)
Graph each line to estimate the solution.
1. y = 3x + 3 (0, 2)
y = -2x + 2
2. y = -3x + 4 No Solution
y = -3x + 2
3. y = 4
y=6
4. y + 3x = 1
No Solution
(1, -2)
2
y  x 3
3
________
5. y = -2x – 1 (1, -3)
x= 1
6. y – 4 = 2x
y = 2x + 6
7. y – 5x = 2
y = 5x + 2
Infinite Solutions
8. 4y = 3x – 2 (-.5, -1)
y = -2x – 2
9. y = 3x + 1
4y = 12x + 4
Infinite Solutions
10. x = -3
x=3
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No Solution
No Solution
Section 6-1: Graphing Systems of Equations (Day 2)
Review Question
What does a solution to a system of equations look like? Use your arms as lines to demonstrate
each possibility.
Point Infinite No Solution
Discussion
Can you look at a system of equations and tell whether it will have 1, infinite, or no solution?
How? Yes, look at the slopes. Different Slopes: 1 solution, Same slopes: no solution, Same
equation: infinite
SWBAT find the solution to a system of equations by graphing
Example 1: How many solutions? Find the solution.
y = 2x + 3
No Solution
y = 2x + 5
Example 2: How many solutions? Find the solution.
y = -3x + 2
1
y = 2x + 5
Example 3: How many solutions? Find the solution.
y = 2x + 5
Infinite Solutions
4y – 8x = 20
You Try!
How many solutions? Then find the solution.
1. y = -4x + 1
1
y = 2x – 3
2. 3y = 2x + 3
No Solution
3. x + 2y = 3
3x – y = -5
1
4. y = 2x – 3
4x = 2y + 6
Infinite Solutions
2
y  x3
_________
3
What did we learn today?
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Section 6-1 Homework (Day 2)
State how many solutions there are going to be then graph each line to find the solution.
1. y = 4x + 1
y = -2x + 1
2. y = -2x + 5
y + 2x = 2
3. x = 2
x=3
4. y + 2x = 1
y
1
x 3
2
________
5. 2y + 4x = -2
y = -2x – 1
6. y = 3x – 1
-3x = y + 2
7. y – 2x = 2
y = 2x + 2
8. 3y = 4x – 2
y = -2x – 2
9. y = x + 1
2y = -2x + 2
10. y = 4
x = -1
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Section 6-1: Graphing Systems of Equations (Day 3)
Review Question
What are the three possibilities for a solution to a system of equations? Use your arms as lines to
demonstrate each possibility. Point Infinite No Solution
Discussion
What is the major issue with solving a system of equations by graphing? It is not precise.
Today, we will be using the graphing calculator to find exact solutions.
SWBAT find the solution to a system of equations by using a graphing calculator
Example 1: How many solutions? 1
y = 4x – 2
y = -2x + 3
Graph to find the solution.
How do you know that your answer is wrong? That point does not work in both equations
Let’s find the exact answer using the graphing calculator.
Example 2: How many solutions? 0
y = -2x + 3
y + 2x = -5
Let’s prove our answer using the graphing calculator.
Example 3: How many solutions? Infinite Solutions
y = 4x + 5
2y – 8x = 10
Let’s confirm our answer using the graphing calculator.
Example 4: How many solutions? 1
y = 8x – 1
3y + 5x = 55
Let’s find the exact answer using the graphing calculator.
Why can’t we see the intersection point? We have to change the window
What did we learn today?
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Section 6-1 In-Class Assignment (Day 3)
Estimate the answer by graphing. Then find the exact answer using the graphing
calculator.
1. y = 4x – 3 (1, 0)
y = -2x + 2 (.83, .33)
2. y = 2x + 6
y – 2x = 1
3. 5y – 4x = 5 (-1, 0)
y = -2x + 2 (-.71, .43)
4. 3y = 6x + 9 Infinite Solutions
y = 2x + 3 Infinite Solutions
No Solution
No Solution
Use the graphing calculator to find the exact answer and sketch the graph.
5. y = -3x + 2 (.6, .2)
y = 2x – 1
6. 5y = 3x – 2
y = -5x – 2
(-.29, -.57)
7. y = 5x – 1
y = 5x + 2
No Solution
8. y = x + 1
3y = 3x + 3
Infinite Solutions
9. y – 4x = 3
y = 4x + 3
Infinite Solutions
10. y   x  23
1
3
y
215
1
x  15
2
(45.6, 7.8)
Section 6-2: Substitution (Day 1)
Review Question
What issue do we have with graphing? It isn’t exact.
Today we will discuss a way to find the exact answer to a system of equations.
Discussion
Solve: 2x + 5 = 11.
How can you check to make sure that ‘3’ is the correct answer? Substitute ‘3’ in for x.
What does substitution mean? Replacing something with something else.
That is what we will be doing today. This allows us to find exact answers to systems of equations. Since
graphing did not.
Solving 2x + 5 = 11 is pretty easy.
Why would solving the following system be difficult?
y = 3x + 5
2x + 4y = 8
There are two equations and two variables. If we could get it down to one equation/one variable, it would
be easy. This is what substitution allows us to do.
SWBAT find the solution to a system of equations by using substitution
Example 1: y = 3x – 2
2x + 3y = 27
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 3x – 2
What does the answer (3, 7) mean? That is the point of intersection
Example 2: x = 3y + 2
6x – 2y = -4
We need to get rid of one variable/equation. We do this by substitution.
What is ‘x’ equal to? x = 3y + 2
What does the answer (-1, -1) mean? That is the point of intersection
Example 3: x + 5y = -3
3x – 2y = 8
What is different about this problem? None of the variables are solved for already
What variable should we solve for? Why? The x in the first equation
What does the answer (2, -1) mean? That is the point of intersection
Summarize
When is it easy to use substitution? When a variable is solved for or can be easily solve for
216
You Try!
1. y = 4x + 1
3x + 2y = 35
(3, 13)
2. x = 3y – 4
2x + 4y = 12
(2, 2)
3. 8x + 2y = 14
3x + y = 6
(1, 3)
4. 2x – y = -4
-3x + y = -9
(13, 30)
What did we learn today?
Section 6-2 Homework (Day 1)
Solve each system of equations using substitution.
1. y = 5x
x + y = 12
(2, 10)
2. y = 3x + 4
3x + 2y = 26
3. x = 4y – 5
2x + 3y = -21
(-9, -1)
4. y = 3x + 2
2x + y = 17
(3, 11)
5. y = 5x + 1
3x + y = -15
(-2, -9)
6. y = 2x + 2
2x – 4y = -26
(3, 8)
7. 3x + 2y = 7
x + 3y = 7
(1, 2)
8. y = 4x – 2
x + 3y = 20
(2, 6)
9. 4x + y = 16
2x + 3y = 18
(3, 4)
10. y = 5x
y = 3x + 2
(1, 5)
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(2, 10)
Section 6-2: Substitution (Day 2)
Review Question
How does substitution help us solve the following system?
y = 3x + 5
2x + 4y = 8
It allows us to eliminate one of the equations/variables.
Discussion
Solve: 2x + 5 = 2x + 7.
What does 5 = 7 mean? There is no solution to this problem.
Solve: 2x + 7 = 2x + 7.
What does 7 = 7 mean? There are infinite solutions to this problem.
SWBAT find the solution to a system of equations by using substitution
Example 1: y = 2x + 3
3x + 3y = 45
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 2x + 3
What does the answer (4, 11) mean? Point of intersection
Example 2: y = 2x + 3
-4x + 2y = 6
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 2x + 3
When does 6 = 6? Always
What does that mean? We have infinite solutions
What kind of lines do we have? They are on top of each other
Example 3: -x + y = 4
-3x + 3y = 10
We need to get rid of one variable/equation. We do this by substitution.
What variable should we solve for? Why? y
When does 12 = 10? Never
What does that mean? There is no solution
What kind of lines do we have? We have parallel lines
Summarize
When is it easy to use substitution? When a variable is solved for or can easily be solved for
218
You Try!
1. y = 3x + 5
4x + 2y = 40
(3, 14)
2. y = 2x + 3
-4x + 2y = 12
No Solution
3. y – 3x = 4
-9x + 3y = 12
Infinite Solutions
4. 4x + y = 11
3x + 2y = 12
(2, 3)
What did we learn today?
Section 6-2 Homework (Day 2)
Solve each system of equations using substitution.
1. y = 3x
(3, 9)
2. y = 4x + 3
2x + 3y = 33
3x + 2y = 28
(2, 11)
3. y = 3x + 2
-3x + y = 10
No Solution
4. 4x + y = 13
3x + 5y = 14
(3, 1)
5. y = 5x + 2
-10x + 2y = 4
Infinite Solutions
6. 3x – 2y = 4
-4x + y = -7
(2, 1)
7. y = 4x – 3
3x + 3y = 51
(4, 13)
8. 2x + y = 10
6x + 3y = 30
Infinite Solutions
9. y = 3x + 1
2x + 3y = 36
(3, 10)
10. -2x + 8y = 8
x – 4y = 10
No Solution
219
Section 6-2: Substitution (Day 3)
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solved for
Discussion
How do you get better at something? Practice
Today will be a day of practice.
SWBAT find the solution to a system of equations by using substitution
Example 1: Let’s make sure we know how to use substitution.
y = 2x + 3
(2, 7)
2x + 3y = 25
Let’s graph to confirm our answer.
You Try!
1. y = 5 – 2x
3y + 3x = 12
(1, 3)
2. y = 4 – 2x
2x – y = 0
(1, 2)
3. x + y = 6
3x + 3y = 3
No Solution
4. 2x + y = 3
4x + 2y = 6
Infinite Solutions
Section 6-2 In-Class Assignment (Day 3)
Solve each system of equations using substitution. Confirm your answer by graphing.
1. y = 3x
2x + 3y = -11
(-1, -3)
2. y = 3
3x + 2y = 9
(1, 3)
3. x = -4
3x + y = -10
(-4, 2)
4. 2x + y = 5
4x + 2y = 2
No Solution
220
5. y = 2x + 1
4x + 2y = 10
(1, 3)
6. 6x – 2y = 5
-12x + 4y = -10
Infinite Solutions
Solve each system of equations using substitution.
7. y = 3x
x + 2y = -21
(-3, -9)
8. x + 5y = 11
3x – 2y = -1
(1, 2)
9. y = 3x + 4
2x + 3y = 34
(2, 10)
10. -2x + 2y = 4
x – 4y = -11
(1, 3)
11. y = 4x – 6
3x + 4y = 33
(3, 6)
12. 2x + y = 7
3x – 2y = 7
(3, 1)
13. y = 3x + 1
2x + 3y = -19
(-2, 5)
14. x + 3y = 14
2x – 4y = -2
(3, 5)
15. x = 2y + 4
2x + 3y = 22
(8, 2)
16. -3x + 2y = -8
x – 4y = 6
(10, 1)
What did we learn today?
221
Section 6-3: Adding/Subtracting (Day 1)
Review Question
What are the three possibilities for a solution to a system of equations? Use your arms as lines to
demonstrate each possibility. Point Infinite No Solution
Discussion
When is it easy to use substitution? When a variable is solved for or can be easily solve for
Why do we substitute something in for a variable? It allows us to get rid of one
variable/equation
How does this help us? We can solve one equation with one variable
Why wouldn’t substitution be good for the following system? When you solve for one of
4x + 5y = 12
variables, the result will be a fraction
4x – 3y = -4
What is something else that we could do? Subtract; it would get rid of one variable/equation
(Remember our goal to get rid of one variable/equation)
Remember the section title;
Remember the Alamo!)
Why are we allowed to add or subtract two equations to each other? Since both sides are equal
to each other, we can add/subtract to both sides.
Just like: 2x + 5 = 11
-5 -5
So, when is it good to use addition/subtraction? When the coefficients are the same
How do you know whether to add or subtract? Same signs: subtract; Different Signs: add
SWBAT find the solution to a system of equations by using addition/subtraction
Example 1: 3x – 2y = 4
(2, 1)
4x + 2y = 10
Why would we use addition not subtraction? Because it eliminates y’s
Example 2: 4x + 5y = 12
(.5, 2)
4x – 3y = -4
Why would we use subtraction not addition? Because it eliminates x’s
Example 3: 2x – 3y = 10
(-1, -4)
2x = y + 2
What is different about this system? The x’s and y’s are not on the same side of the equation
222
Summarize
When is it easy to use addition/subtraction? When the coefficients are the same
How do you know whether to add or subtract? Same signs: subtract; Different Signs: add
You Try!
1. 4x – 5y = 10
2x + 5y = 20
(5, 2)
2. 3x + 5y = -16
3x – 2y = -2
(-2, -2)
3. y = 4x + 2
3x + 4y = 27
(1, 6)
4. -6x + 2y = 2
6x = 3y – 3
(0, 1)
5. 4x + 2y = 16
4x + 2y = 10
No Solution
6.
(5, 1)
3x + y = 16
6x – 3y = 27
What did we learn today?
Section 6-3 Homework (Day 1)
Use addition, subtraction, or substitution to solve each of the following system of
equations.
1. 3x + 2y = 22
3x – 2y = 14
(6, 2)
3. 3x – 5y = -35
2x – 5y = -30
5. 4x = 7 – 5y
8x = 9 – 5y
7. x – 3y = 7
x + 2y = 2
(-5, 4)
(.5, 1)
(4, -1)
9. 4x + y = 12 (2, 4)
3x + 3y = 18
223
2. 3x + 2y = 30
y = 2x + 1
(4, 9)
4. 5x + 2y = 12 (2, 1)
-5x + 4y = -6
6. x = 6y + 11 (23, 2)
2x + 3y = 52
8. 3x + 5y = 12 All Reals
3x + 5y = 12
10. 2x + 3y = 5 (4, -1)
5x + 4y = 16
Section 6-3: Adding/Subtracting (Day 2)
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solve for
When is it easy to use addition/subtraction? When the coefficients are the same
Discussion
What method should we use for problem #10 on the homework? Substitution
Why does this stink? It involves fractions.
How do you get better at something? Practice
Today will be a day of practice.
SWBAT find the solution to a system of equations by using addition/subtraction
Example 1: 5x – 4y = 8
(4, 3)
4x + 4y = 28
Why would we use addition not subtraction? It will eliminate the y’s
Example 2: 5x + 5y = -5
(1, -2)
5x – 3y = 11
Why would we use subtraction not addition? It will eliminate the x’s
You Try!
1. 4x – 7y = -13
2x + 7y = 25
(2, 3)
2. 3x + 4y = -9
3x = 2y + 9
(1, -3)
3. y = -2x – 3
4x + 2y = -6
Infinite Solutions
4. 3x + 2y = 11
3x + 2y = 8
No Solution
What did we learn today?
224
Section 6-3 Homework (Day 2)
Use addition, subtraction, or substitution to solve each of the following system of
equations.
1. 5x + 4y = 14
5x + 2y = 12
(2, 1)
2. 3x + 6y = 21 (1, 3)
-3x+ 4y = 9
3. 5x + 2y = 6 (4, -7)
9x + 2y = 22
4. y = -3x + 2 (0, 2)
3x + 2y = 4
5. 2x – 3y = -11
x + 3y = 8
6. 6x + 5y = 8 No Solution
6x + 5y = - 2
(-1, 3)
7. x = 3y + 7 Infinite Solutions
3x – 9y = 21
8. 3x – 4y = -5
3x = -2y + 7
9. 2x + 3y = 1
x + 5y = 4
10. 4x – 5y = 2 (3, 2)
6x + 5y = 28
225
(-1, 1)
(1, 2)
Section 6-3: Adding/Subtracting (Day 3)
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solve for
When is it easy to use addition/subtraction? When the coefficients are the same
Discussion
What is our goal when we are trying to solve a system of equations? Get rid of one variable
How does this help us? We can solve an equation with one variable
SWBAT solve a word problem that involves a system of equations
Example 1: Find two numbers whose sum is 64 and difference is 42.
x + y = 64
x – y = 42
2x = 106
x = 53
y = 11
Example 2: Cable costs $50 for installation and $100/month. Satellite costs $200 for installation
and $70/month. What month will the cost be the same?
C = 50 + 100m
C = 200 + 70m
0 = -150 + 30m
150 = 30m
5=m
What does 5 months represent? The month where it costs the same for both.
How could this help you decide on which company to go with?
What did we learn today?
Section 6-3 Homework (Day 3)
Use addition, subtraction, or substitution to solve each of the following system of
equations.
1. 2x + 2y = -2
3x – 2y = 12
(2, -3)
2. 4x – 2y = -1 (-1, -1.5)
-4x + 4y = -2
226
3. 6x + 5y = 4 (-1, 2)
6x – 7y = - 20
4. x = 3y + 7 (4, -1)
3x + 4y = 8
5. 2x – 3y = 12
4x + 3y = 24
(6, 0)
6. 3x + 2y = 10
3x + 2y = -8
No Solution
7. 4x + 2y = 10
2x + y = 5
Infinite Solutions
8. 8x + y = 10 (1, 2)
2x – 5y = -8
Write a system of equations. Then solve.
9. The sum of two numbers is 70 and their difference is 24. Find the two numbers. (23, 47)
10. Twice one number added to another number is 18. Four times the first number minus the
other number is 12. Find the numbers. (5, 8)
11. Two angles are supplementary. The measure of one angle is 10 more than three times the
other. Find the measure of each angle. (42.5, 137.5)
12. Johnny is older than Jimmy. The difference of their ages is 12 and the sum of their ages is
50. Find the age of each. (31, 19)
13. The sum of the digits of a two digit number is 12. The difference of the digits is 2. Find the
number if the units digit is larger than the tens digit. 5 and 7
14. A store sells Cd’s and Dvd’s. The Cd’s cost $4 and the Dvd’s cost $7. The store sold a total
of 272 items and took in $1694. How many of each was sold? 202, 70
227
Section 6-4: Multiplication (Day 1)
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solve for
When is it easy to use addition/subtraction? When the coefficients are the same
Discussion
What is our goal when we are trying to solve a system of equations? Get rid of one variable
How does this help us? We can solve an equation with one variable
What method would you use to solve the following system of equations?
2x + 3y = 5
5x + 4y = 16
(4, -1)
Why wouldn’t substitution be good? It will not eliminate any of the variables
Why wouldn’t add/subtract be good? It will not eliminate any of the variables
We need something else.
SWBAT solve a system of equations using multiplication
Example 1: 9x + 8y = 10
18x + 3y = 33
(2, -1)
What is something else we could do? Multiplication
Remember we are trying to get rid of one of the variables.
Example 2: 2x + 3y = 5
(4, -1)
5x + 4y = 16
How is this problem different from the previous one? You need to multiply both equations for
the variables to be eliminated
Hmmmm
When should we use multiplication? When the coefficients are different
How about division? It is the same as multiplying
Dividing by 2 is the same as multiplying by 1/2
You Try!
1. 2x + 3y = 8
4x + 5y = 14
(1, 2)
2. 4x + 5y = -7
6x – 3y = 21
(2, - 3)
3. x = 5y + 7
2y – x = 8
(-18, -5)
4. 3x – 4y = 12
3x – 4y = -14
No Solution
228
5. -x + y = -15
(13, -2)
-4y = x – 5
6. 2x – 3y = 1
(2, 1)
5x + 5y = 15
What did we learn today?
Section 6-4 Homework (Day 1)
Use addition, subtraction, substitution, or multiplication to solve each of the
following system of equations.
1. 5x + 4y = 19
2x + 2y = 8
(3, 1)
2. 2x + 6y = 28 (2, 4)
3x + 4y = 22
3. 5x + 2y = 4 (2, -3)
8x + 2y = 10
4. y = 4x + 3 (1, 7)
4x + 2y = 18
5. 2x – 3y = -16
3x + 3y = 6
6. 3x + 2y = -11
6x + 5y = -23
(-2, 4)
(-3, -1)
7. 4x = 4y – 4 (1, 2)
3x – 9y = -15
8. 3x – 4y = 10 Infinite Solutions
9x – 12y = 30
9. 2x + 3y = 1
x + 5y = 4
10. 2x – 5y = -2
6x + 5y = 34
(4, 2)
11. x = 2y + 3 (11, 4)
3x + 2y = 41
12. 3x – 4y = 10
3x = 4y + 5
No Solution
13. 2x + 3y = -1
2x + 5y = 1
14. 3x – 2y = 7 (3, 1)
5x + 3y = 18
229
(-1, 1)
(-2, 1)
Section 6-4: Multiplication (Day 2)
Review Question
When do we use multiplication to solve a system of equations?
When the coefficients are different
Why is it important to know all of the different methods?
Makes it easier; must know + and – to use multiplication
Discussion
Which method should you use?
1. 4x + 6y = 12
3x – 2y = 13
Multiplication
2. y = 3x + 2
2x – 5y = 12
Substitution
3. 2x + 5y = -11
-2x – 2y = 11
Addition
SWBAT solve a system of equations using multiplication
Example 1: 2x + 3y = 5
-5x – 2y = -18
(4, -1)
You Try!
1. 2x + 4y = 10
3x – 2y = 7
(3, 1)
2. y = 3x + 2
4x – 5y = 12
(-2, -4)
3. 2x + 5y = -6
-4x – 2y = 12
(-3, 0)
4. 4x + 3y = 15
2x – 3y = 3
(3, 1)
230
What did we learn today?
Section 6-4 In-Class Assignment (Day 2)
1. x = 5y – 6
x + 2y = 8
(4, 2)
2. -2x + y = 5
2x + 3y = 3
(-3/2, 2)
3. 2x + 3y = 6
4x + 6y = 18
No Solution
4. 3x + 2y = 7
4x + 7y = 18
(1, 2)
5. 3x – 2y = -7
y=x+4
(1, 5)
6. 3x = 2 – 7y
14y = -6x + 4
All Reals
7. 4x + 6y = -10
8x – 3y = -5
(-1, -1)
8. 8x – 7y = 5
3x – 5y = 9
(-2, -3)
9. 6x + 3y = -9
2x – 3y = -7
(-2, 1)
10. 2x = 2y + 6
5x – 2y = 18
(4, 1)
231
Section 6-4: Multiplication (Day 3)
Review Question
When do we use multiplication to solve a system of equations?
When the coefficients are different
Why is it important to know all of the different methods?
Makes it easier; must know + and – to use multiplication
Discussion
Today we are going to solve some word problems that require multiplication to solve. We solved
some word problems that required adding and subtracting.
What is difficult about these problems? Setting up the initial system
SWBAT solve a word problem that involves multiplication to solve
Example 1: Johnny has $2.55 in nickels and dimes. He has a total of 31 coins. How many of
each coin does he have?
.05n + .10d = 2.55
n + d = 31
dimes: 20, nickels: 11
Example 2: It costs $8 for adults and $5 for kids at the movie theatre. The theatre sold 107
tickets and collected a total of $670. How many of each ticket did they sell?
8a + 5k = 670
a + k = 107
Adults: 45, Kids: 62
When would someone have to write an equation based on a real world problem?
Computer programming; cash registers
What did we learn today?
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Section 6-4 In-Class Assignment (Day 3)
1. y = 3x – 2
x + 2y = 17
(3, 7)
2. 4x + 6y = 0
4x + 3y = -6
(-3, 2)
3. 4x + 5y = 6
6x – 7y = -20
(-1, 2)
4. y = 4x – 3
2x – y = 1
(1, 1)
5. 2x – 5y = -2
4x + 5y = 26
(4, 2)
6. 2x – 4y = 8
x – 2y = 3
No Solution
7. Timmy made 145 baskets this year. Some were 2 pointers, some were 3 pointers. He scored a
total of 335 points. How many 2 and 3 pointers did he make? 2 pointers: 100, 3 pointers: 45
8. Amy is 5 years older than Ben. Three times Amy’s age added to six times Ben’s age is 42.
How old are Amy and Ben? Amy: 8, Ben: 3
9. The school cafeteria sold a total of 140 lunches. Some of the lunches were pizza and some
were spaghetti. Pizza costs $1.50 and spaghetti costs $2. If the cafeteria collected $239, how
many of each lunch did they sell? Pizza: 82, Spaghetti: 58
10. Two numbers add up to 82. Three times the bigger number minus two times the smaller
number is 131. What are the two numbers? 59, 23
233
Section 6-4: Multiplication (Day 4)
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solve for
When is it easy to use addition/subtraction? When the coefficients are the same
When is it easy to use multiplication? When the coefficients are different
Discussion
If you truly understand something, then you can talk freely about it. Specifically, you should be
able to come up with your own explanations about the topic. This is what we will be doing today.
SWBAT make up a word problem that requires a system of equations to solve
You are going to make up your own problems today. In order to make up your own problems,
you will have to work backwards in order to ensure your answer will make sense.
The first type of problem that you will make up will involve buying two different items. First,
figure out what the two items are going to be. Next, make up how many of each item you are
going to buy. Finally, make up how much each item costs.
“Two Items”- Just Thinking
Jimmy bought 2 items (shirt, pants).
I’m thinking 8 shirts, 4 pants.
The shirts are $12. The pants are $20.
Therefore, he bought a total of 12 items for a total cost of $176. ($96 shirts, $80 pants)
This will lead us to our actual problem…
“Two Items”- Actual Problem
Jimmy bought some shirts @ $12 each. He bought some pants @ $20 each. He bought a total of
12 items. He spent a total $176. How many of each did he buy?
12s + 20p = 176
s + p = 12
s = 8, p = 4
The second type of problem that you will make up will involve two different numbers. First,
figure out what the two numbers are going to be. Next, figure out two different ways the numbers
are related.
“Two #’s”- Just Thinking
The two numbers I am thinking of are 12 and 26. Therefore, my problem would be: The two
numbers add up to 38. If you double the first number then add two you will get the second
number. This will lead us to our actual problem…
234
“Two #’s”- Actual Problem
Two numbers add up to 38. If you double the first number then add two you will get the second
number. What are the two numbers?
x + y = 38
2x + 2 = y
x = 12, y = 26
Activity
Make up and solve two word problems. The first problem will be “two items” and the second
problem will be “two numbers”.
For each problem:
 Write a paragraph explaining the problem.
 Write an appropriate system of equations.
 Write a complete solution.
* You can use HW problems to help you.
What did we learn today?
235
Section 6-5: Graphing Systems of Inequalities (Day 1)
Review Question
When is it easy to use substitution? When a variable is solved for or can be easily solved for
When is it easy to use addition/subtraction? When the coefficients are the same
When is it easy to use multiplication? When the coefficients are different
Discussion
How do you find the solution to a system of equations by graphing? Find the intersection point.
What does that point represent? The point that will “work” in both equations.
How do you find the solution to a system of inequalities by graphing?
Let’s come back to that in a minute…
What did the graph of y > 3x + 1 look like? Line with a shaded region.
What does the solution look like? The shaded region.
What do you think the solution to a system of inequalities looks like?
SWBAT graph a system of inequalities to find the solution set
Example 1: Graph: y 
2
x 3
3
y < -2x + 1
Start at (0, -3), up 2 over 3
Start at (0, 1), down 2 over 1
What does the answer mean? Any point in the shaded region will “work”.
Example 2: Graph: 2x + y > 4
y < -2x – 1
Start at (0, 4), down 2 over 1
Start at (0, -1), down 2 over 1
Is it possible to have parallel lines and the answer not be No Solution? How? Yes, if and only if
their shaded regions intersect
Example 3: Graph: -3x + y < 4
-3y < 3x + 6
Start at (0, 4), up 3 over 1
Start at (0, -2), down 1 over 1
You Try!
1. y < -4x + 1
y > 2x – 4
Start at (0, 1), down 4 over 1
Start at (0, -4), up 2 over 1
2. y + 3x > 2
y<3
Start at (0, 2), down 3 over 1
Horizontal line at 3
3. y > -4
x<3
Horizontal line at -4
Vertical line at 3
236
4. y < x – 1
-2y < -4x – 2
Start at (0, -1), up 1 over 1
Start at (0, 1), up 2 over 1
How do you know that the lines aren’t parallel? Different slopes
What did we learn today?
Section 6-5 Homework (Day 1)
Solve each system of inequalities by graphing.
1. y > 4x + 1
y < -2x + 1
2. y > -3x + 5
y < -3x + 1
3. y > 4
y>6
4. y + 3x < 1
y
1
x3
3
________
5. y > -2x – 1
x>1
6. y – 4 < 2x
y > 2x + 6
7. y – 3x > 2
y < 5x + 2
8. 4y > 3x – 2
y > -2x – 2
9. y > 3x + 1
4y < 12x + 4
10. x > -3
x<3
237
Section 6-5: Graphing Systems of Inequalities (Day 2)
Review Question
How do we know what the answer to a system of inequalities is?
It is the intersection of each inequality when graphed.
Discussion
Yesterday, we graphed a system of inequalities. Today, I am going to give you a graph of a
system of inequalities and see if you can write the actual system. For example, what system of
inequalities is represented by the graph below?
y > -2
x > -3
SWBAT write a system of inequalities based on a graph
Example 1: What system of inequalities is represented by the graph below?
y < 1/2x + 1
y > -1/2x – 1
Example 2: What system of inequalities is represented by the graph below?
y < 1x + 2
y > 1x – 2
238
What did we learn today?
Section 6-5 In-Class Assignment (Day 2)
Write a system of inequalities based on the graph.
1.
2.
3.
4.
Solve each system of inequalities by graphing.
5. y > 3x + 3
y < -4x + 4
6. y > -2x
y < -2x + 4
7. y > -1
x>2
8. y + 2x < 4
y
1
x3
4
________
9. y > -4x – 3
x>4
10. y – 1 < 3x
y > 3x + 7
11. y – x > 2
y < 4x + 2
12. 2y < -4x – 4
y > -2x – 2
239
Section 6-5: Graphing Systems of Inequalities (Day 3)
Review Question
What are the possibilities for a solution to a system of equations?
A point, No Solution, infinite solutions
What are the possibilities for a solution to a system of inequalities?
Region, No Solution, Line
How can the solution be a line? Look at problem #12
y>x
y<x
Discussion
Yesterday, we graphed systems of inequalities by hand. Today, we are going to graph them using
the graphing calculators. Why? Easier, Need to know how to use them in the future
SWBAT graph a system of inequalities using a graphing calculator to find the solution set
Example 1: y > 3x + 7
y + x < -4
Start at (0, 7), up 3 over 1
Start at (0, -4), down 1 over 1
Let’s graph it by hand first.
Now, let’s check it with the graphing calculators.
Example 2: y > -2x + 3
y + 2x < -8
Start at (0, 3), down 2 over 1
Start at (0, -8), down 2 over 1
No Solution
Let’s graph using the graphing calculator.
Example 3: y > 3x + 7
3y + 5x < -8
Start at (0, 7), up 3 over 1
Start at (0, -8/3), down 5 over 3
Let’s graph using the graphing calculator. Make sure to put the 2nd inequality into “y =” form.
Example 4: y > 4
x < -2
Horizontal line at 4
Vertical line at -2
Let’s graph it by hand first.
Now, let’s check it with the graphing calculators.
What issue do we have? x < -2 doesn’t go into the graphing calculator
What did we learn today?
240
Section 6-5 In-Class Assignment (Day 3)
Solve each system of inequalities by graphing by hand then confirm your answer on the
graphing calculator.
1. y > 3x + 2
y < -2x + 4
2. y > -3x + 7
y < -3x + 2
3. y > 2
y > -2
4. y + 2x < 3
y
1
x3
4
________
Solve each system of inequalities by sketching the solution from the graphing calculator.
5. y > -2x – 1
y > 4x + 3
6. y – 5 < 2x
y < 2x + 1
7. y – 3x > 2
y < 5x + 2
8. 5y > 3x – 3
y > -2x – 2
9. y > 3x + 2
4y < 12x + 8
10. y > -5
x<2
241
Unit 6 Review
Review Question
How do you know what the solution to a system of inequalities is?
The intersection of the shaded regions.
SWBAT review for the Unit 6 Test
Discussion
1. How do you study for a test? The students either flip through their notebooks at home or do not
study at all. So today we are going to study in class.
2. How should you study for a test? The students should start by listing the topics.
3. What topics are on the test? List them on the board
- Graphing
- Substitution
- Adding/Subtracting
- Multiplication
- Graphing Systems of Inequalities
4. How could you study these topics? Do practice problems
Practice Problems
Have the students do the following problems. They can do them on the dry erase boards or as an
assignment. Have students place dry erase boards on the chalk trough. Have one of the groups explain
their solution.
Graph each system of equations. Then determine whether the system has one solution, no solution,
or infinitely many solutions. If the system has one solution, name it.
1. y = -x + 2
y = 2x + 7
(-2, 4)
3. y + 2x = -1
y – 4 = -2x
2. 3x + y = 5
2y – 10 = -6x
Infinite
No Solution
Use substitution or elimination to solve each system of equations.
4. y = 7 – x
x – y = -3
6. 2x + 5y = 12
x – 6y = -11
(2, 5)
(1, 2)
5. x + y = 8
x–y=2
7. 8x – 6y = 14
6x – 9y = 15
(5, 3)
(1, -1)
242
8. 5x – y = 1
y = -3x + 1
(1/4, 1/4)
Solve each system of inequalities by graphing.
9. y < 3
y > -x + 2
10. x < 2y
2x + 3y < 7
11. x > y + 1
2x + y > -4
Write a system of equations. Then solve.
12. The difference between the length and width of a rectangle is 7 cm. Find the dimensions of the
rectangle if its perimeter is 50 cm. l = 16, w = 9
13. Joey sold 30 peaches from his fruit stand for a total of $750. He sold small ones for 20 cents each
and large ones for 35 cents each. How many of each kind did he sell? s = 20, l = 10
What did we learn today?
243
UNIT 6 CUMULATIVE REVIEW
SWBAT do a cumulative review
Discussion
What does cumulative mean?
All of the material up to this point.
Does anyone remember what the first six chapters were about? Let’s figure it out together.
1. Pre-Algebra
2. Solving Linear Equations
3. Functions
4. Linear Equations
5. Inequalities
6. Systems
Things to Remember:
1. Reinforce test taking strategies: guess/check, eliminate possibilities, work backwards, and estimating.
2. Reinforce the importance of retaining information from previous units.
3. Reinforce connections being made among units.
In-Class Assignment
1. What set of numbers does -5 belong?
a. counting
b. whole
c. integers
d. irrationals
2. 4 + 2 = 2 + 4 is an example of what property?
a. Commutative
b. Associative
c. Distributive
d. Identity
3. -8.2 + (-4.2) =
a. -12.4
b. -3.8
c. 12.4
d. -9.8
b. 10/12
c. 7/24
d. 2/3
5. (-2.5)(4.7) =
a. -9.88
b. -7.2
c. -11.75
d. -5.9
6. 5.18 ÷ 1.4 =
a. 4.8
b. 3.2
c. 6.52
d. 3.7
1
6
4. 1 
2

4
a. 20/12
244
7.  2
1 10


2 3
a. – 2/12
b. -1/4
c. -3/4
d. 8/9
33
a. 3
b. 9
c. 12
d. 27
b. 29
c. 220.5
d. 87
b. 3 7
c. 8
d. 7 3
11. 18 – 24 ÷ 12 + 3
a. 15
b. 16
c. 19
d. 20
12. 3x + 4y – 8x + 6y
a. 11x +10y
b. 5x + 2y
c. 5x + 10y
d. -5x + 10y
13. 2x + 2 = 14
a. 6
b. -6
c. 8
d. -8
14. 2x + 8 = 5x + 23
a. -5
b. -6
c. No Solution
d. Reals
15. 2(x – 3) – 6x = -6 – 4x
a. 5
b. 6
c. No Solution
d. Reals
c. y = -5x – 4a
d. y = -5x – 4a/2
8.
441 =
9.
a. 21
63 =
10.
a. 31.5
16. Solve for y: 4a + 3y = -5x
a. y = 5x – 4a
b. y 
 5x  4a
3
17. Which of the following is a solution to y = 3x + 5 given a domain of {-3, 0, 1}
a. (0, 5)
b. (1, 2)
c. (-3, -1)
d. (-3, 7)
18. Which equation is not a linear equation?
a. y = -3x + 2
245
b.
x
y
4
c. y = 5
d. y = x2 + 1
19. Which equation is not a function?
a. y = 3x + 7
b. y = 5
c. x = -5
d. y = 1/2x + 2
20. If g(x) = 4x – 3, find g(3).
a. 4
b. 5
c. 8
d. 9
21. Write an equation for the following relation: (2, 10) (6, 8) (10, 6)
a. y = -2x
b. y = 4x + 12
1
2
c. y   x  11
22. Write an equation of a line that passes through the points (3, 6) and (4, 8).
a. y = x
b. y = -2x
c. y = 2x + 12
23. Write an equation of a line that is perpendicular to y 
a. y = x
b. y = -3x
d. y = 2x – 11
d. y = 2x
1
x  2 and passes thru (-1, 3).
3
c. y = 3x + 6
d. y = 3x
24. Write an equation of a line that is parallel to y + 2x = -2 and passes thru (3, -2).
a. y = -2x + 4
b. y = -2x
c. y = -2x + 8
d. y = 2x
25. Write an equation of a line that is perpendicular to x = -3 and passes thru the point (2, -4).
a. y = 2
b. y = -4
c. y = 2x
d. y = 4
26. Which of the following is a graph of: y = 2x – 5.
a.
b.
c.
d.
27. Which of the following is a graph of: y = 3
a.
b.
c.
d.
246
28. What is the x-intercept of the line y = 4x + 8?
a. 4
b. 8
c. -2
d. 2
29. Which of the following is a graph of: y < 2x + 3.
a.
b.
c.
d.
b. x < 30
c. x > 30
d. x > -30
31. |2x + 8| > 14
a. x > 3 or x < -11
b. x > 3 and x < -11
c. x < -11
d. x > 3
32. |4x + 1 | > -2
a. x > -3/4
b. x < 1/2
c. No Solution
d. Reals
c. (3/2, 1/2)
d. (-3, 1)
c. (4, 2)
d. (-3, 7)
c. (1, 1)
d. (-3, 5)
30.
x
 3  12
2
a. x < -30
33. Solve the following system of equations.
y=x+2
2x + 3y = 11
a. (0, 2)
b. (1, 3)
34. Solve the following system of equations.
3x – y = 10
7x – 2y = 24
a. (0, 5)
b. (6, 2)
35. Solve the following system of equations.
2x – 6y = 4
2x – 6y = 10
a. No Solution
247
b. Infinite
Standardized Test Review
1. Anna burned 15 calories per minute running x minutes and 20 calories per minute hiking for y minutes.
She spent a total of 60 minutes running and hiking and burned 1000 calories. The system of equations
shown below can be used to determine how much time Anna spent of each exercise.
15x + 20y = 1000
x + y = 60
What is the value of x, the minutes Anna spent running?
a. 10
b. 20
c. 30
d. 40
2. Which system is graphed below?
a. 2x + y = -3
y = -2x – 1
b. 4x + y = -3
-x + y = -3
c. 2x + y = 3
6x + 3y = 9
d. 2x + y = 3
y = -2x – 1
c. (1, 5)
d. (-10, 6)
3. Solve the system: 3x + 4y = 23
5x + 4y + 25
a. (3, 2)
b. (5, 2)
4. Several books are on sale at a bookstore. Fiction books cost $4, while non-fiction books cost $6. One
day last week 80 books were sold. The total amount of sales was $400. The system of equations shown
below can be used to determine how many of each type of book were sold. Let x stand for the number of
fiction books and y stand for the number of non-fiction books.
4x + 6y = 400
x + y = 80
Which of the following statements is true?
a.
b.
c.
d.
There were 30 non-fiction books sold.
Fiction books cost more than non-fiction books.
Exactly twice as many fiction books were sold than non-fiction books.
They sold the same amount of non-fiction and fiction books.
248
5. The solution set to a system of linear inequalities is graphed below.
Which system of 2 linear inequalities has the solution set shown in the graph?
a. x > 1
y>x+3
b. y > 1
y > -x + 3
c. y > 1
y < -x + 3
d. x > 1
y > -x + 3
6. The following problems require a detailed explanation of the solution. This should include all
calculations and explanations.
The following problem involves systems of equations.
a. What are the three possible solutions to a system of equations? (Explain using sentences and
pictures)
b. Make up a system of equations for each one of these possibilities. (Don’t solve them.)
c. Why isn’t it possible to have a system of linear equations that has two solutions?
249