Unit 6 Systems - Aliquippa School District Home Manual62.pdf · SWBAT find the solution to a system...

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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

210

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 Same m’s;

Different intercepts

Infinite Infinite 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


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.


y = 2x – 3

y = 2x + 3

No Solution

What does the answer of No Solution mean? No numbers will work in both equations.

Section 6-1: Graphing Systems of Equations (Day 1)

211


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 (.5, .5) 2. y = 4x – 1 No Solution

y = -3x + 1 y – 4x = 2

3. y = 4x + 1 Infinite Solutions 4. y + 3x = 1 (1, -3)

3y = 12x + 3

________

5. 3y – 8 = 4x (2, 5) 6. y = 5 No Solution

x = 2 y = -2

What did we learn today?

Graph each line to estimate the solution.

1. y = 3x + 3 (0, 2) 2. y = -3x + 4 No Solution

y = -2x + 2 y = -3x + 2

3. y = 4 No Solution 4. y + 3x = 1 (1, -2)

y = 6

________

5. y = -2x – 1 (1, -3) 6. y – 4 = 2x No Solution

x = 1 y = 2x + 6

7. y – 5x = 2 Infinite Solutions 8. 4y = 3x – 2 (-.5, -1)

y = 5x + 2 y = -2x – 2

9. y = 3x + 1 Infinite Solutions 10. x = -3 No Solution

4y = 12x + 4 x = 3

Section 6-1 Homework (Day 1)

33

2 xy

33

2 xy

212

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


y = -3x + 2 1

y = 2x + 5


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 1

3x – y = -5

4. y = 2x – 3 Infinite Solutions

4x = 2y + 6



33

2 xy

213

State how many solutions there are going to be then graph each line to find the solution.

1. y = 4x + 1 2. y = -2x + 5

y = -2x + 1 y + 2x = 2

3. x = 2 4. y + 2x = 1

x = 3

________

5. 2y + 4x = -2 6. y = 3x – 1

y = -2x – 1 -3x = y + 2

7. y – 2x = 2 8. 3y = 4x – 2

y = 2x + 2 y = -2x – 2

9. y = x + 1 10. y = 4

2y = -2x + 2 x = -1


32

1 xy

214

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.


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.


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



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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)

3. 5y – 4x = 5 (-1, 0)

y = -2x + 2 (-.71, .43)

2. y = 2x + 6 No Solution

y – 2x = 1 No Solution

4. 3y = 6x + 9 Infinite Solutions

y = 2x + 3 Infinite Solutions

Use the graphing calculator to find the exact answer and sketch the graph.

5. y = -3x + 2 (.6, .2)

y = 2x – 1

7. y = 5x – 1 No Solution

y = 5x + 2

9. y – 4x = 3 Infinite Solutions

y = 4x + 3

6. 5y = 3x – 2 (-.29, -.57)

y = -5x – 2

8. y = x + 1 Infinite Solutions

3y = 3x + 3

10. 233

1 xy (45.6, 7.8)

152

1 xy

Section 6-1 In-Class Assignment (Day 3)

216

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


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

Section 6-2: Substitution (Day 1)

217

You Try!

1. y = 4x + 1 (3, 13)

3x + 2y = 35

2. x = 3y – 4 (2, 2)

2x + 4y = 12

3. 8x + 2y = 14 (1, 3)

3x + y = 6

4. 2x – y = -4 (13, 30)

-3x + y = -9


Solve each system of equations using substitution.

1. y = 5x (2, 10) 2. y = 3x + 4 (2, 10)

x + y = 12 3x + 2y = 26

3. x = 4y – 5 (-9, -1) 4. y = 3x + 2 (3, 11)

2x + 3y = -21 2x + y = 17

5. y = 5x + 1 (-2, -9) 6. y = 2x + 2 (3, 8)

3x + y = -15 2x – 4y = -26

7. 3x + 2y = 7 (1, 2) 8. y = 4x – 2 (2, 6)

x + 3y = 7 x + 3y = 20

9. 4x + y = 16 (3, 4) 10. y = 5x (1, 5)

2x + 3y = 18 y = 3x + 2


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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.


Example 1: y = 2x + 3

3x + 3y = 45


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


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


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


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You Try! 1. y = 3x + 5 (3, 14)

4x + 2y = 40

2. y = 2x + 3 No Solution

-4x + 2y = 12

3. y – 3x = 4 Infinite Solutions

-9x + 3y = 12

4. 4x + y = 11 (2, 3)

3x + 2y = 12



1. y = 3x (3, 9) 2. y = 4x + 3 (2, 11)

2x + 3y = 33 3x + 2y = 28

3. y = 3x + 2 No Solution 4. 4x + y = 13 (3, 1)

-3x + y = 10 3x + 5y = 14

5. y = 5x + 2 Infinite Solutions 6. 3x – 2y = 4 (2, 1)

-10x + 2y = 4 -4x + y = -7

7. y = 4x – 3 (4, 13) 8. 2x + y = 10 Infinite Solutions

3x + 3y = 51 6x + 3y = 30

9. y = 3x + 1 (3, 10) 10. -2x + 8y = 8 No Solution

2x + 3y = 36 x – 4y = 10


220

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.


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 (1, 3)

3y + 3x = 12

2. y = 4 – 2x (1, 2)

2x – y = 0

3. x + y = 6 No Solution

3x + 3y = 3

4. 2x + y = 3 Infinite Solutions

4x + 2y = 6

Solve each system of equations using substitution. Confirm your answer by graphing.

1. y = 3x (-1, -3) 2. y = 3 (1, 3)

2x + 3y = -11 3x + 2y = 9

3. x = -4 (-4, 2) 4. 2x + y = 5 No Solution

3x + y = -10 4x + 2y = 2



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5. y = 2x + 1 (1, 3) 6. 6x – 2y = 5 Infinite Solutions

4x + 2y = 10 -12x + 4y = -10


7. y = 3x (-3, -9) 8. x + 5y = 11 (1, 2)

x + 2y = -21 3x – 2y = -1

9. y = 3x + 4 (2, 10) 10. -2x + 2y = 4 (1, 3)

2x + 3y = 34 x – 4y = -11

11. y = 4x – 6 (3, 6) 12. 2x + y = 7 (3, 1)

3x + 4y = 33 3x – 2y = 7

13. y = 3x + 1 (-2, 5) 14. x + 3y = 14 (3, 5)

2x + 3y = -19 2x – 4y = -2

15. x = 2y + 4 (8, 2) 16. -3x + 2y = -8 (10, 1)

2x + 3y = 22 x – 4y = 6


222

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

Section 6-3: Adding/Subtracting (Day 1)

223

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 (5, 2) 2. 3x + 5y = -16 (-2, -2)

2x + 5y = 20 3x – 2y = -2

3. y = 4x + 2 (1, 6) 4. -6x + 2y = 2 (0, 1)

3x + 4y = 27 6x = 3y – 3

5. 4x + 2y = 16 No Solution 6. 3x + y = 16 (5, 1)

4x + 2y = 10 6x – 3y = 27


Use addition, subtraction, or substitution to solve each of the following system of

equations.

1. 3x + 2y = 22 (6, 2) 2. 3x + 2y = 30 (4, 9)

3x – 2y = 14 y = 2x + 1

3. 3x – 5y = -35 (-5, 4) 4. 5x + 2y = 12 (2, 1)

2x – 5y = -30 -5x + 4y = -6

5. 4x = 7 – 5y (.5, 1) 6. x = 6y + 11 (23, 2)

8x = 9 – 5y 2x + 3y = 52

7. x – 3y = 7 (4, -1) 8. 3x + 5y = 12 All Reals

x + 2y = 2 3x + 5y = 12

9. 4x + y = 12 (2, 4) 10. 2x + 3y = 5 (4, -1)

3x + 3y = 18 5x + 4y = 16


224

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 (2, 3)

2x + 7y = 25

2. 3x + 4y = -9 (1, -3)

3x = 2y + 9

3. y = -2x – 3 Infinite Solutions

4x + 2y = -6

4. 3x + 2y = 11 No Solution

3x + 2y = 8



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equations.

1. 5x + 4y = 14 (2, 1) 2. 3x + 6y = 21 (1, 3)

5x + 2y = 12 -3x+ 4y = 9

3. 5x + 2y = 6 (4, -7) 4. y = -3x + 2 (0, 2)

9x + 2y = 22 3x + 2y = 4

5. 2x – 3y = -11 (-1, 3) 6. 6x + 5y = 8 No Solution

x + 3y = 8 6x + 5y = - 2

7. x = 3y + 7 Infinite Solutions 8. 3x – 4y = -5 (1, 2)

3x – 9y = 21 3x = -2y + 7

9. 2x + 3y = 1 (-1, 1) 10. 4x – 5y = 2 (3, 2)

x + 5y = 4 6x + 5y = 28


226

Review Question



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?



equations.

1. 2x + 2y = -2 (2, -3)

3x – 2y = 12

2. 4x – 2y = -1 (-1, -1.5)

-4x + 4y = -2



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3. 6x + 5y = 4 (-1, 2)

6x – 7y = - 20

4. x = 3y + 7 (4, -1)

3x + 4y = 8

5. 2x – 3y = 12 (6, 0)

4x + 3y = 24

6. 3x + 2y = 10 No Solution

3x + 2y = -8

7. 4x + 2y = 10 Infinite Solutions

2x + y = 5

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

228

Review Question



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 (4, -1)

5x + 4y = 16

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 (2, -1)

18x + 3y = 33

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 (1, 2) 2. 4x + 5y = -7 (2, - 3)

4x + 5y = 14 6x – 3y = 21

3. x = 5y + 7 (-18, -5) 4. 3x – 4y = 12 No Solution

2y – x = 8 3x – 4y = -14

Section 6-4: Multiplication (Day 1)

229

5. -x + y = -15 (13, -2) 6. 2x – 3y = 1 (2, 1)

-4y = x – 5 5x + 5y = 15


Use addition, subtraction, substitution, or multiplication to solve each of the

following system of equations.

1. 5x + 4y = 19 (3, 1) 2. 2x + 6y = 28 (2, 4)

2x + 2y = 8 3x + 4y = 22

3. 5x + 2y = 4 (2, -3) 4. y = 4x + 3 (1, 7)

8x + 2y = 10 4x + 2y = 18

5. 2x – 3y = -16 (-2, 4) 6. 3x + 2y = -11 (-3, -1)

3x + 3y = 6 6x + 5y = -23

7. 4x = 4y – 4 (1, 2) 8. 3x – 4y = 10 Infinite Solutions

3x – 9y = -15 9x – 12y = 30

9. 2x + 3y = 1 (-1, 1) 10. 2x – 5y = -2 (4, 2)

x + 5y = 4 6x + 5y = 34

11. x = 2y + 3 (11, 4) 12. 3x – 4y = 10 No Solution

3x + 2y = 41 3x = 4y + 5

13. 2x + 3y = -1 (-2, 1) 14. 3x – 2y = 7 (3, 1)

2x + 5y = 1 5x + 3y = 18


230

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 (4, -1)

-5x – 2y = -18

You Try!

1. 2x + 4y = 10 (3, 1)

3x – 2y = 7

2. y = 3x + 2 (-2, -4)

4x – 5y = 12

3. 2x + 5y = -6 (-3, 0)

-4x – 2y = 12

4. 4x + 3y = 15 (3, 1)

2x – 3y = 3


231


1. x = 5y – 6 (4, 2) 2. -2x + y = 5 (-3/2, 2)

x + 2y = 8 2x + 3y = 3

3. 2x + 3y = 6 No Solution 4. 3x + 2y = 7 (1, 2)

4x + 6y = 18 4x + 7y = 18

5. 3x – 2y = -7 (1, 5) 6. 3x = 2 – 7y All Reals

y = x + 4 14y = -6x + 4

7. 4x + 6y = -10 (-1, -1) 8. 8x – 7y = 5 (-2, -3)

8x – 3y = -5 3x – 5y = 9

9. 6x + 3y = -9 (-2, 1) 10. 2x = 2y + 6 (4, 1)

2x – 3y = -7 5x – 2y = 18


232

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



233

1. y = 3x – 2 (3, 7) 2. 4x + 6y = 0 (-3, 2)

x + 2y = 17 4x + 3y = -6

3. 4x + 5y = 6 (-1, 2) 4. y = 4x – 3 (1, 1)

6x – 7y = -20 2x – y = 1

5. 2x – 5y = -2 (4, 2) 6. 2x – 4y = 8 No Solution

4x + 5y = 26 x – 2y = 3

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


234

Review Question



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…


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“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.


236

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 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: Start at (0, -3), up 2 over 3

y < -2x + 1 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 Start at (0, 4), down 2 over 1

y < -2x – 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 Start at (0, 4), up 3 over 1

-3y < 3x + 6 Start at (0, -2), down 1 over 1

You Try!

1. y < -4x + 1 Start at (0, 1), down 4 over 1

y > 2x – 4 Start at (0, -4), up 2 over 1

2. y + 3x > 2 Start at (0, 2), down 3 over 1

y < 3 Horizontal line at 3

3. y > -4 Horizontal line at -4

x < 3 Vertical line at 3

Section 6-5: Graphing Systems of Inequalities (Day 1)

33

2 xy

237

4. y < x – 1 Start at (0, -1), up 1 over 1

-2y < -4x – 2 Start at (0, 1), up 2 over 1

How do you know that the lines aren’t parallel? Different slopes


Solve each system of inequalities by graphing.

1. y > 4x + 1 2. y > -3x + 5

y < -2x + 1 y < -3x + 1

3. y > 4 4. y + 3x < 1

y > 6

________

5. y > -2x – 1 6. y – 4 < 2x

x > 1 y > 2x + 6

7. y – 3x > 2 8. 4y > 3x – 2

y < 5x + 2 y > -2x – 2

9. y > 3x + 1 10. x > -3

4y < 12x + 4 x < 3


33

1 xy

238

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


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Write a system of inequalities based on the graph.

1. 2.

3. 4.


5. y > 3x + 3 6. y > -2x

y < -4x + 4 y < -2x + 4

7. y > -1 8. y + 2x < 4

x > 2

________

9. y > -4x – 3 10. y – 1 < 3x

x > 4 y > 3x + 7

11. y – x > 2 12. 2y < -4x – 4

y < 4x + 2 y > -2x – 2


34

1 xy

240

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 Start at (0, 7), up 3 over 1

y + x < -4 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 Start at (0, 3), down 2 over 1

y + 2x < -8 Start at (0, -8), down 2 over 1

No Solution

Let’s graph using the graphing calculator.

Example 3: y > 3x + 7 Start at (0, 7), up 3 over 1

3y + 5x < -8 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 Horizontal line at 4

x < -2 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



241

Solve each system of inequalities by graphing by hand then confirm your answer on the

graphing calculator.

1. y > 3x + 2 2. y > -3x + 7

y < -2x + 4 y < -3x + 2

3. y > 2 4. y + 2x < 3

y > -2

________

Solve each system of inequalities by sketching the solution from the graphing calculator.

5. y > -2x – 1 6. y – 5 < 2x

y > 4x + 3 y < 2x + 1

7. y – 3x > 2 8. 5y > 3x – 3

y < 5x + 2 y > -2x – 2

9. y > 3x + 2 10. y > -5

4y < 12x + 8 x < 2


34

1 xy

242

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 (-2, 4) 2. 3x + y = 5 Infinite

y = 2x + 7 2y – 10 = -6x

3. y + 2x = -1 No Solution

y – 4 = -2x

Use substitution or elimination to solve each system of equations.

4. y = 7 – x (2, 5) 5. x + y = 8 (5, 3)

x – y = -3 x – y = 2

6. 2x + 5y = 12 (1, 2) 7. 8x – 6y = 14 (1, -1)

x – 6y = -11 6x – 9y = 15

Unit 6 Review

243

8. 5x – y = 1 (1/4, 1/4)

y = -3x + 1


9. y < 3 10. x < 2y

y > -x + 2 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


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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.

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

4. 4

2

6

11

a. 20/12 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

In-Class Assignment

UNIT 6 CUMULATIVE REVIEW

245

7. 3

10

2

12

a. – 2/12 b. -1/4 c. -3/4 d. 8/9

8. 33

a. 3 b. 9 c. 12 d. 27

9. 441 =

a. 21 b. 29 c. 220.5 d. 87

10. 63 =

a. 31.5 b. 73 c. 8 d. 37

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

16. Solve for y: 4a + 3y = -5x

a. y = 5x – 4a b. 3

45 axy

c. y = -5x – 4a d. y = -5x – 4a/2

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 b. yx

4 c. y = 5 d. y = x

2 + 1

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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 c. 112

1 xy d. y = 2x – 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 d. y = 2x

23. Write an equation of a line that is perpendicular to 23

1 xy and passes thru (-1, 3).

a. y = x b. y = -3x 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.

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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.

30. 1232

x

a. x < -30 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

33. Solve the following system of equations.

y = x + 2

2x + 3y = 11

a. (0, 2) b. (1, 3) c. (3/2, 1/2) d. (-3, 1)


3x – y = 10

7x – 2y = 24

a. (0, 5) b. (6, 2) c. (4, 2) d. (-3, 7)


2x – 6y = 4

2x – 6y = 10

a. No Solution b. Infinite c. (1, 1) d. (-3, 5)

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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 b. 4x + y = -3 c. 2x + y = 3 d. 2x + y = 3

y = -2x – 1 -x + y = -3 6x + 3y = 9 y = -2x – 1

3. Solve the system: 3x + 4y = 23

5x + 4y + 25

a. (3, 2) b. (5, 2) c. (1, 5) d. (-10, 6)

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. There were 30 non-fiction books sold.

b. Fiction books cost more than non-fiction books.

c. Exactly twice as many fiction books were sold than non-fiction books.

d. They sold the same amount of non-fiction and fiction books.

Standardized Test Review

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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 b. y > 1 c. y > 1 d. x > 1

y > x + 3 y > -x + 3 y < -x + 3 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?

Unit 6 Systems - Aliquippa School District Home Manual62.pdf · SWBAT find the solution to a system...

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