Aliquippa School District Home
|Unit 5 – Systems |Length of section |
|5-1 Graphing |4 days |
|5-2 Substitution |4 days |
|Test Review |1 day |
|Test |1 day |
|Cumulative Review |1 day |
|Unit Project |1 day |
| |
|Total days in Unit 5 - Systems = 12 days |
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)
How do you graph y = 3x + 1? Start at 1; then go up 3 and over 1.
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.
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) is the solution because that is the intersection point.
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 = -5
[pic]
The solution looks like it would be (2, 2).
How do you know that your answer is incorrect? (2, 2) doesn’t “work” in either equation
The correct solution is (1.8, 2.1).
Example 3: Graph each line to find the solution.
y = 3x + 2
y = 5
(1, 5) is the solution because that is the intersection point.
How do you know that your answer is correct? (2, 1) “works” in both equations
You Try!
Graph each line to estimate the solution.
1. y = 4x + 1 (1, 4) 2. y = 2x – 4 (1, -2)
y = -3x + 6 y = -3x + 2
3. y + 2x = 1 (1, -1) 4. y = 4x + 2 (2, 10)
[pic] x = 2
What did we learn today?
Graph each line to estimate the solution.
1. y = 3x + 1 (1, 3) 2. y = -3x + 4 (1, 2)
y = -2x + 5 y = -8x + 8
3. y = 4x – 2 (2, 6) 4. y + 3x = 1 (1, -2)
y = 6 [pic]
5. y = -3x – 4 (1, -7) 6. y = x + 4 (1, 5)
x = 1 y = -3x + 6
7. y – 5x = 2 (-1, -2) 8. y = 3x – 2 (0, -2)
[pic] y = -2x – 2
9. y = -3x + 5 (1, 3) 10. y = 2 (3, 2)
y = 2x + 2 x = 3
Review Question
What is a system of equations? 2 or more equations
What is a solution to a system of equations? Point of intersection
Discussion
What is the answer to the following system of equations? Why? Empty Set. The two lines don’t intersect.
SWBAT find the solution to a system of equations by graphing
Example 1: Graph each line to estimate the solution.
y = 2x – 3
y = 2x + 3
Empty Set
What does the answer of empty set mean? No numbers will work in both equations.
Example 2: Graph each line to estimate the solution.
__________
Empty Set
Example 3: Graph each line to estimate 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.
Example 4: Graph each line to estimate the solution.
y = 2
y = 5
Empty Set
You Try!
Graph each line to estimate the solution.
1. y = 2x + 3 2. y = 4x – 1
y = -3x + 1 y = 4x + 2
(0, 2) Empty Set
3. y = 4x + 1 4. y + 3x = 1
3y = 12x + 3
Infinite Solutions
________
(1, -2)
5. y = 5x + 2 6. y = 5
x = 2 y = -2
(2, 12) Infinite Solutions
What did we learn today?
Graph each line to estimate the solution.
1. y = 3x + 3 2. y = -3x + 4
y = -2x + 2 y = -3x + 2
(0, 2) Empty Set
3. y = 4 4. y + 3x = 1
y = 6
Empty Set
________
(1, -2)
5. y = -2x – 1 6. y = 2x + 4
x = 1 y = 2x + 6
(1, -3) Empty Set
7. y – 5x = 2 8. y = 3x – 2
y = 5x + 2 y = -2x – 2
Infinite Solutions (0, -2)
9. y = 3x + 1 10. x = -3
4y = 12x + 4 x = 3
Infinite Solutions Empty Set
Review Question
What are the possibilities for a solution to a system of equations? Use your arms as lines to demonstrate each possibility. Point, Infinite, Empty Set
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
y = 2x + 1 y = 3x + 4 y = 4x + 2
y = 2x – 5 y = -2x + 2 y = 4x + 2
No solution 1 solution Infinite solutions
SWBAT find the solution to a system of equations by graphing
Example 1: How many solutions? Estimate the solution.
y = 2x + 3
y = 2x + 5
0; No Solutions
Example 2: How many solutions? Estimate the solution.
y = -3x + 2
y = 2x + 5
1; (-1, 4)
Example 3: How many solutions? Estimate the solution.
y = 2
4y = 8
Infinite; Infinite Solutions
You Try!
How many solutions? Then estimate the solution.
1. y = -3x + 1 2. y = x + 2
y = -3x – 4 y = -2x – 1
0; No Solution 1; (-1, 1)
3. x = 3 4. y = 2x – 3
y = 5 y – 2x = -3
1; (3, 5) Infinite; Infinite Solutions
What did we learn today?
State how many solutions there are going to be. Then graph each line to estimate 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. y = -2x – 1 6. y = 3x – 1
y = -2x – 1 y = 3x + 2
7. y – 2x = 2 8. y = 4x – 2
y = 2x + 2 y = -2x – 2
9. y = x + 1 10. y = 4
2y = 2x + 2 x = -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
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. (1, 1)
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.
1. Press the “y =” button. Enter each equation.
2. Press graph.
3. Press 2nd, then trace.
4. Scroll down to 5: Intersect. Press enter.
5. Press enter 3 times.
(.83, 1.33)
Example 2: How many solutions? 0; Empty Set
y = -2x + 3
y + 2x = -5
Let’s confirm our answer using the graphing calculator.
(See instructions from above.)
Example 3: How many solutions? Infinite Solutions
y = 4x + 5
2y = 8x + 10
Let’s confirm our answer using the graphing calculator.
(See instructions from above.)
What did we learn today?
Estimate the answer by graphing. Then find the exact answer using the graphing calculator.
1. y = 4x – 3 2. y = 2x + 6
y = -2x + 2 y = 2x + 1
3. y – 4x = 5 4. 3y = 6x + 9
y = -2x + 2 y = 2x + 3
Use the graphing calculator to find the exact answer and sketch the graph.
5. y = -3x + 2 6. y = 3x – 2
y = 2x – 1 y = -5x – 2
7. y = 5x – 1 8. y = x + 1
y = 5x + 2 3y = 3x + 3
9. y – 4x = 3 10. [pic]
y = 4x + 3 [pic]
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 it back into the equation
What does substitution mean? Putting something in place of 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 (2x + 5 = 11), it would be easy. This is what substitution allows us to do.
SWBAT solve a system of equations by using substitution
Example 1: y = 2
x + y = 6
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? y = 2
So we substitute ‘2’ in for ‘y’ in the second equation. When we do this, the second equation becomes:
x + 2 = 6. Now solve. Notice how substitution got rid of an equation and a variable.
- 2 - 2
x = 4
The final answer is (4, 2).
What does the answer (4, 2) mean? That is the point of intersection.
Example 2: y = 3x + 2
3x + y = 14
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? 3x + 2
So we substitute ‘3x + 2’ in for ‘y’ in the second equation. When we do this, the second equation becomes:
3x + (3x + 2) = 14. Now solve. Notice how substitution got rid of an equation and a variable.
6x + 2 = 14
- 2 - 2
6x = 12
6 6
x = 2
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 3(2) + 2
y = 6 + 2
y = 8
The final answer is (2, 8).
What does the answer (2, 8) mean? That is the point of intersection.
Example 3: y = 2x
x + 2y = 10
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? 2x
So we substitute ‘2x’ in for ‘y’ in the second equation. When we do this, the second equation becomes:
x + 2(2x) = 10. Now solve. Notice how substitution got rid of an equation and a variable.
x + 4x = 10
5x = 10
5 5
x = 2
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 2(2)
y = 4
The final answer is (2, 4).
What does the answer (2, 4) mean? That is the point of intersection.
Example 4: y = 2x + 2
2x + 3y = 30
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? 2x + 2
So we substitute ‘2x + 2’ in for ‘y’ in the second equation. When we do this, the second equation becomes:
2x + 3(2x + 2) = 30. Now solve. Notice how substitution got rid of an equation and a variable.
2x + 6x + 6 = 30
8x + 6 = 30
- 6 - 6
8x = 24
8 8
x = 3
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 2(3) + 2
y = 6 + 2
y = 8
The final answer is (3, 8).
What does the answer (3, 8) mean? That is the point of intersection.
Summarize
When is it easy to use substitution? When a variable is solved for
You Try!
1. y = 4 2. y = 2x + 3
x + y = 9 4x + y = 21
(5, 4) (3, 9)
3. y = 3x 4. y = 4x + 1
2x + 4y = 28 3x + 2y = 35
(2, 6) (3, 13)
What did we learn today?
Solve each system of equations using substitution.
1. y = 5 2. y = 3
x + y = 11 2x + 3y = 17
(6, 5) (4, 3)
3. y = 5x 4. y = 3x + 2
3x + y = 32 2x + y = 17
(4, 20) (3, 11)
5. y = 5 6. y = 5x
y = 2x – 1 y = 3x + 2
(3, 5) (1, 5)
7. y = 2x + 2 8. y = 4x – 2
2x + 4y = 58 x + 3y = 20
(5, 12) (2, 6)
9. y = 8 10. y = 3x
2x + 3y = 24 (0, 8) 3x + 2y = 27
(0, 8) (3, 9)
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 + y = 23
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? 2x + 3
So we substitute ‘2x + 3’ in for ‘y’ in the second equation. When we do this, the second equation becomes:
3x + 2x + 3 = 23. Now solve. Notice how substitution got rid of an equation and a variable.
5x + 3 = 23
- 3 - 3
5x = 20
5 5
x = 4
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 2(4) + 3
y = 8 + 3
y = 11
The final answer is (4, 11).
What does the answer (4, 11) mean? It is the point where the two lines will intersect. It will “work” in both equations.
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? 2x + 3
So we substitute ‘2x + 3’ in for ‘y’ in the second equation. When we do this, the second equation becomes:
-4x + 2(2x + 3) = 6. Now solve. Notice how substitution got rid of an equation and a variable.
-4x + 4x + 6 = 6
6 = 6
When does 6 = 6? Always
What does that mean? Infinite amount of solutions
What kind of lines do we have? They are the same line.
Example 3: y = x + 4
-3x + 3y = 10
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? x + 4
So we substitute ‘x + 4’ in for ‘y’ in the second equation. When we do this, the second equation becomes:
-3x + 3(x + 4) = 10. Now solve. Notice how substitution got rid of an equation and a variable.
-3x + 3x + 12 = 10
12 = 10
When does 12 = 10? Never
What does that mean? No solution will work. Empty Set
What kind of lines do we have? Parallel
Summarize
When is it easy to use substitution? When a variable is solved for
You Try!
1. y = 3x
x + 2y = 28 (4, 12)
2. y = 2x + 3
-4x + 2y = 12 Empty Set
3. y = 3x + 4
-9x + 3y = 12 Infinite Solutions
4. y = 4x + 2
3x + 2y = 26 (2, 10)
What did we learn today?
Solve each system of equations using substitution.
1. y = 3x 2. y = 4x
2x + 3y = 33 3x + y = 7
(3, 9) (1, 4)
3. y = 3x + 2 4. y = 5x + 2
-3x + y = 10 -10x + 2y = 4
Empty Set Infinite Solutions
5. y = 5x + 2 6. y = 5
3x + 2y = 17 y = 11
(1, 7) Empty Set
7. y = 3x + 1 8. y = 4x – 2
2x + 3y = 36 3x + 2y = 29
(3, 10) (3, 10)
9. y = 3x + 2 10. y = 6x + 3
-12x + 4y = 8 -6x + y = 6
Infinite Solutions Empty Set
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, Empty Set
Discussion
How do you get better at something? Practice
Therefore, we are going to practice solving systems using substitution today.
We are going to have many days like this during the school year. In order for you to be successful, you need to take advantage of the time and ask questions from your classmates and teachers.
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
2x + 3y = 25
We need to get rid of one variable/equation. We do this by substitution.
What is ‘y’ equal to? 2x + 3
So we substitute ‘2x + 3’ in for ‘y’ in the second equation. When we do this, the second equation becomes:
2x + 3(2x + 3) = 25. Now solve. Notice how substitution got rid of an equation and a variable.
2x + 6x + 9 = 25
8x + 9 = 25
- 9 - 9
8x = 16
8 8
x = 2
Now substitute the ‘x’ back into either of the first two equations to get the ‘y’ value.
y = 2(2) + 3
y = 4 + 3
y = 7
The final answer is (2, 7).
What does the answer (2, 7) mean? It is the point where the two lines will intersect.
How do you know that the answer is correct? It will “work” in both equations.
You Try!
1. y = 2x – 5 2. y = 3x + 2
3x + 3y = 12 -15x + 5y = 0
(3, 1) Empty Set
3. y = -x + 1 4. y = 3x + 4
3x + 3y = 3 4x + 2y = 28
Infinite Solutions (2, 10)
What did we learn today?
Graph each line to find the solution.
1. y = 3x + 2 2. x = 2
y = -2x + 4 y = 3
(.4, 3.2) (2, 3)
3. y = -2x + 5 4. [pic]
y = -2x + 2 [pic]
Empty Set Infinite Solutions
Solve each system of equations using substitution.
5. y = 2x 6. x = 3
2x + 4y = 40 x = 5
(4, 8) Empty Set
7. y = 4x 8. y = 3x + 2
5x + y = 27 2x + 4y = 8
(3, 12) (0, 2)
9. y = 4x + 2 10. y = 2x – 3
-4x + y = 10 3x + 2y = 22
Empty Set (4, 5)
11. y = 4x + 2 12. y = 5x + 2
-8x + 2y = 4 -10x + 2y = 8
Infinite Solutions Empty Set
13. y = 4x + 3 14. y = 4x + 3
3x + 2y = 61 -4x + y = 3
(5, 23) Infinite Solutions
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, Empty Set
Discussion
Back in Unit 3, we were writing equations based on real life situations. For example: a phone costs $250 plus $80 per month: C = $250 + $80m.
In Unit 4, we wrote equations like this in slope intercept form: y = 80m + 250.
Notice the $250 represents the starting point (y-intercept) and the $80 represents the amount the bill increases each month (slope).
Today we are going to write equations based on real life examples. The difference is that we are going to look at two different equations at the same time. This is called a system of equations.
SWBAT write a system of equations based on a real life example
SWBAT find the solution to a system of equations by using substitution
Example 1: 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
We need to get rid of one variable/equation. We do this by substitution.
What is ‘C’ equal to? 50 + 100m
So we substitute ‘50 + 100m’ in for ‘C’ in the second equation. When we do this, the second equation becomes:
50 + 100m = 200 + 70m (Now solve. Notice how substitution got rid of an equation and a variable.)
- 70m - 70m
50 + 30m = 200
- 50 - 50
30m = 150
30 30
m = 5
What does 5 months represent? The month where it costs the same for both gyms.
How could this help you decide on which company to go with? Depending on how long you are going to keep your cable.
Example 2: The temperature on Thursday was 50° F and decreased 2° F per hour. The temperature on Friday was 65° F and decreased 5° F per hour. After how many hours was the temperature the same?
T = 50 – 2h
T = 65 – 5h
We need to get rid of one variable/equation. We do this by substitution.
What is ‘T’ equal to? 50 – 2h
So we substitute ‘50 – 2h’ in for ‘T’ in the second equation. When we do this, the second equation becomes:
50 – 2h = 65 – 5h (Now solve. Notice how substitution got rid of an equation and a variable.)
+ 2h + 2h
50 = 65 – 3h
- 65 - 65
-15 = -3h
-3 -3
5 = h
What did we learn today?
Graph each line to find the solution.
1. y = 4x + 2 2. x = -4
y = -2x + 8 y = 5
(1, 6) (-4, 5)
3. y = 5x + 1 4. y – 4x = 3
y = 5x – 3 y = 4x + 3
Empty Set Infinite Solutions
Solve each system of equations using substitution.
5. y = 3x 6. y = 4
2x + 4y = 28 y = -3
(2, 6) Empty Set
7. y = x 8. y = 4x + 1
5x + y = 18 2x + 4y = 40
(3, 3) (2, 9)
9. y = 5x + 2 10. y = 5x – 3
-5x + y = 10 3x + 2y = 7
Empty Set (1, 2)
11. y = 5x + 2 12. y = 7x + 2
-10x + 2y = 4 -14x + 2y = 1
Infinite Solutions Empty Set
13. y = 8x + 3 14. y = x + 4
3x + 2y = 6 -x + y = 4
(0, 3) Infinite Solutions
15. Write a system of equations. Then solve. Timmy has $70 in his bank account and saves $20 per week. Jimmy has $40 in his bank account and saves $30 per week. After how many weeks will they have the same amount of money? 3 weeks
16. Write a system of equations. Then solve. Microsoft stock is $32 per share every month. It went up $2 per month. Apple stock is $95 per share. It went up $2 per share every month. After how many weeks with the stocks be the same price? Never
Review Question
How does substitution help us solve a system of equations?
It allows us to eliminate one of the equations/variables.
SWBAT study for our Unit 5 test
Discussion
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.
How should you study for a test? The students should start by listing the topics.
What topics are on the test? List them on the board
- Graphing Systems
- Substitution
How could you study these topics? Do practice problems; study the topics that you are weak on
You Try!
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. 3x + y = 5
y = 2x + 2 y = -3x + 5
1; (0, 2) Infinite Solutions
3. y + 2x = -1 4. y = -3
y – 4 = -2x x = 4
Empty Set (4, -3)
Solve each system of equations using substitution.
5. y = x 6. y = 3
2x + 4y = 30 y = 5
(5, 5) Empty Set
7. y = 5x 8. y = 2x + 4
5x + y = 40 2x + 4y = 26
(4, 20) (1, 6)
9. y = 3x + 2 10. y = 3x – 1
-3x + y = 10 3x + 2y = 16
Empty Set (2, 5)
11. y = 3x + 2 12. y = 4x + 5
-6x + 2y = 4 -12x + 3y = 9
Infinite Solutions Empty Set
13. y = x + 5 14. y = 5x + 1
3x + 2y = 35 -5x + y = 1
(5, 10) Infinite Solutions
15. After you do the review problems, pick out one or two topics that you are weak on and find three problems from your notes or homework and do them.
What did we learn today?
[pic]
1. Anna burned 15 calories per minute running x minutes and 10 calories per minute hiking for y minutes. She spent a total of 60 minutes running and hiking and burned 700 calories. The system of equations shown below can be used to determine how much time Anna spent of each exercise.
15x + 10y = 700
x = 60 – y
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. y = -3x + 2 b. y = 3x + 2 c. y = -3x + 2 d. y = 2x + 2
y = -2x – 1 y = 3x – 1 y = -3x – 1 y = -2x – 1
3. Solve the system: y = 2x + 3
5x + 2y = 33
a. (3, 9) b. (9, 3) c. (3, 6) d. (-10, 6)
4. Several books are on sale at a bookstore. Fiction books cost $3, while nonfiction books cost $5. One day last week 80 books were sold. The total amount of sales was $300. 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 nonfiction books.
3x + 5y = 300
x + y = 80
Which of the following statements is true?
a. There were 30 nonfiction books sold.
b. Fiction books cost more than nonfiction books.
c. Exactly twice as many fiction books were sold than nonfiction books.
d. They sold more nonfiction books than fiction books.
5. Which system of 2 linear equations is shown in the graph below?
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 problem requires a detailed explanation of the solution. This should include all calculations and explanations.
a. Explain the three possibilities for solutions to a system of equations.
b. Is there 1 solution, no solution, or infinite solutions to each of the following systems of equations. Explain how you arrived at your answer. Then graph each system to confirm your answer.
c. Solve the following system of equations using substitution.
y = 2x + 5
4x + 2y = 26
[pic]
SWBAT do a cumulative review
Discussion
What does cumulative mean?
All of the material up to this point.
Our goal is to remember as much mathematics as we can by the end of the year. The best way to do this is to take time and review after each unit. So today we will take time and look back on the first four units.
Does anyone remember what the first five units were about? Let’s figure it out together.
1. Problem Solving
2. Numbers/Operations
3. Pre-Algebra
4. Algebra
5. 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.
[pic]
1. What is the biggest two digit number that is divisible by 2?
a. 100 b. 1000 c. 99 d. 98
2. What is the most popular brand of basketball shoes?
a. Nike b. Crocs c. Uggs d. Vans
3. What is the next term: 1, 2, 4, 7, ___?
a. 11 b. 12 c. 15 d. 17
4. What is the next term: 1, 3, 2, 4, ___?
a. 3 b. 4 c. 5 d. 6
5. Tommy had $95. He spent $8.75 at Wendy’s and $19.99 on a t-shirt. About how much money does he have left?
a. $50 b. $55 c. $65 d. $75
6. What set(s) of numbers does ‘-3.2’ belong?
a. C, W, I, R b. W, I, R c. I, R d. R
7. What number is the smallest 25%, .22, 28%, [pic]?
a. 25% b. .22 c. 28% d. 2/10
8. -7 – 10 =
a. 3 b. -3 c. 17 d. -17
9. (3.2)(-1.5) =
a. -4.6 b. -4.7 c.-4.8 d. -4.9
10. Which of the following is equal to (5)3?
a. 15 b. 25 c. 125 d. 135
11. Which of the following is equal to [pic]?
a. 1/49 b. -1/7 c. -7 d. 7
12. Which of the following is equal to [pic]?
a. 20 b. 27 c. 29 d. 212
13. Which of the following is equal to[pic]?
a. 37 b. 33 c. 27 d. 684.5
14. Which of the following is equal to[pic]?
a. 1 b. 4 c. 5 d. 6
15. Which of the following is equal to 5.2 x 103?
a. .0052 b. 52 c. 520 d. 5200
16. Which of the following is equal to 3.1 x 10-4?
a. .00031 b. 31000 c. .031 d. 31
17. Which of the following is 15 in scientific notation?
a. 1.5 b. 1.5 x 101 c. 15 x 103 d. 1.5 x 103
18. Which of the following is .0246 in scientific notation?
a. 2.46 x 102 b. 2.46 x 10-5 c. 2.46 x 10-2 d. 2.46 x 103
19. 48 – (7 + 3) + 12
a. 50 b. 3 c. 26 d. -17
20. 4(3x – 4)
a. 7x + 4 b. 12x – 16 c. 12x + 8 d. 12x
21. [pic]
a. All Reals b. 24 c. 4 d. -24
22. 6x + 7 = 4x + 2x + 5
a. Empty Set b. All Reals c. 4 d. 5
23. 4(2x + 4) = x + 30
a. Empty Set b. All Reals c. -2 d. 2
24. Johnny has $100. He makes $8.50/hour. How many hours will it take for him to save $168?
a. 10 b. 8 c. 6 d. 2
25. Solve y = 2x + 4; given a domain of {-2, 0, 4}.
a. (-2, 0) (0, 4) (4, 12) b. (-2, 8) (0, 4) (4, 8) c. (-2, -10) (0, -4) (4, 0) d. (-10, -2) (4, 1) (1, 3)
26. Which point is a solution to the following equation: y = 4x + 2?
a. (2, -1) b. (0, 5) c. (9, 2) d. (1, 6)
27. Which equation is not a linear equation?
a. 3x2 + 3y = 5 b. 3x + y = 1 c. x = 2 d. [pic]
28. Write an equation for the following relation: (2, 10) (4, 7) (6, 4)
a. [pic] b. y = 4x + 10 c. [pic] d. [pic]
29. y = 2x + 4
a. b. c. d.
30. x = 3
a. b. c. d.
31. Which of the following is a function?
a. (1,4) (2,5) (3,6) (1,-5) b. (1,4) (2,5) (3,6) (2,3) c. (1,4) (2,5) (3,6) d. (1,2) (2,3) (1,4)
32. Which of the following is a function?
a. b. c. d.
33. Write an equation of a line that has a slope of 3 and a y-intercept of 5.
a. y = 3x + 5 b. y = 5x + 3 c. y = 3x d. y = 5x
34. Write an equation of a line that contains the points (6, 5) and (0, 3).
a. [pic] b. y = 4x + 3 c. [pic] d. [pic]
35. A scatter plot of number of songs on your phone and amount of space left would be what type of relationship?
a. Positive b. Negative c. Scattered d. Weathered
36. The following system of equations will have how many solutions?
y = 2x + 2
y = 2x + 5
a. 0 b. 1 c. 2 d. Infinite
37. The following system of equations will have how many solutions?
y = 3x + 2
y = 2x + 1
a. 0 b. 1 c. 2 d. Infinite
38. Solve the following system of equations.
y = 2x + 3
2x + 3y = 17
a. (0, 2) b. (1, 5) c. (3/2, 1/2) d. (-3, 1)
This problem set is intended to challenge the students and encourage students to apply a deep understanding of problem–solving skills.
Solve each system of equations. There are methods other than substitution to solve a system. You can do some research as to what they are. These other methods will help you greatly with this assignment.
1. y = 3x 2. x + 5y = 11
x + 2y = -21 3x – 2y = -1
3. y = 3x + 4 4. -2x + 2y = 4
2x + 3y = 34 x – 4y = -11
5. y = 3x – 2 6. 4x + 6y = 0
x + 2y = 17 4x + 3y = -6
7. 4x + 5y = 6 8. y = 4x – 3
6x – 7y = -20 2x – y = 1
9. 2x – 5y = -2 10. 2x – 4y = 8
4x + 5y = 26 x – 2y = 3
Working backwards Project
Notice that most of the systems that you solved this unit had answers that were integers. Your teacher did this on purpose. The way to ensure that the answer will “work out” is to work backwards. This is the type of skill that we learned back in Unit 1. In this project, you will be making up systems that “work out” as well.
1. Make up a system of equations that will give you an answer of (1, 1). Then explain the process by which you came up with this system.
2. Make up a system of equations that will give you an answer of (2, 2). Then make up a different system of equations that will give you an answer of (2, 2).
3. Make up a system of equations that will give you an answer of (-4, 5).
4. Make up a system of equations that will give you an answer of empty set.
5. Make up a system of equations that will give you an answer of infinite solutions.
6. Explain how working backwards can help you solve other math problems.
-----------------------
Section 5-1: Graphing (Day 1) (CCSS: 8.EE.8.a, 8.EE.8.b, 8.EE.8.c)
Section 5-1 Homework (Day 1)
Section 5-1: Graphing (Day 2) (CCSS: 8.EE.8.a, 8.EE.8.b, 8.EE.8.c)
[pic]
[pic]
[pic]
Section 5-1 Homework (Day 2)
[pic]
Section 5-1: Graphing (Day 3) (CCSS: 8.EE.8.a, 8.EE.8.b, 8.EE.8.c)
Section 5-1 Homework (Day 3)
[pic]
Section 5-1: Graphing (Day 4) (CCSS: 8.EE.8.a, 8.EE.8.b, 8.EE.8.c)
Section 5-1 In-Class Assignment (Day 4)
Section 5-2: Substitution (Day 1) (CCSS: 8.EE.8.b, 8.EE.8.c)
Section 5-2 Homework (Day 1)
Section 5-2: Substitution (Day 2) (CCSS: 8.EE.8.b, 8.EE.8.c)
Section 5-2 Homework (Day 2)
Section 5-2: Substitution (Day 3) (CCSS: 8.EE.8.b, 8.EE.8.c)
Section 5-2 In-Class Assignment (Day 3)
Section 5-2: Substitution (Day 4) (CCSS: 8.EE.8.b, 8.EE.8.c)
Section 5-2 In-Class Assignment (Day 4)
Unit 5 Review
Standardized Test Review
UNIT 5 CUMULATIVE REVIEW
In-Class Assignment
Unit 5 Hand-In Problems
Unit 5 Project
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