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Pre-AP Algebra II – Chapter 1 Test Review Standards/Goals:

A.1.a.: I can identify properties of real numbers and use them and the correct order of operations to simplify expressions.

A.1.b.: I can multiply monomials and binomials. A.1.c.: I can solve single-step and multistep equations and inequalities in one variable. A.1.e./A.CED.2. /A.REI.6./A.CED.3.::

o I can solve systems of linear equations using various methods, including substitution, elimination with addition and elimination with multiplication.

o I can create a system of equation to represent a real-life situation. o I can solve systems of equations for both an exact answer and an approximation. o I can interpret solutions to systems of equations as being inconsistent, independent or

dependent (one solution, no solution or infinitely many solutions). A.1.h.: I can find the distance and midpoint between two points in the coordinate plane.I.1.a./ N.VM.8.:

I can add, subtract, and multiply matrices. D.1.c./A.CED.2.: I can solve systems of THREE linear equations using various methods, including

substitution and the use of matrices. I.1.b.: I can use addition, subtraction, and multiplication of matrices to solve real-world problems.

o N.VM.6.: I can use matrices to manipulate data. o N.VM.7.: I can multiply matrices by scalars to produce new matrices. o N.VM.8.: I can add, subtract and multiply matrices with appropriate dimensions. o N.VM.10.: I can understand that the zero and identity matrices play a role in matrix addition

and multiplication similar to the role of 0 and 1. I.1.c.: I can calculate the determinant of 2 x 2 and 3 x 3 matrices. I.1.d.: I can find the inverse of a 2 x 2 matrix. I.1.e. /A.REI.8: : I can solve a system of linear equations using a single matrix equation and inverses

and determinants. I.1.f.: I can use technology to perform operations on matrices, find determinants, and find inverses. S.MD.6.: I can use determine the mean, median, and mode for a data set. I can identify outliers from a data set.

S.MD.7.: I can determine measures of ‘spread’ such as range, quartiles and the inter quartile range.

#1. Consider the following matrix to answer the questions.𝐴 =

[ 1 74 −95 23−8 93 12]

a. What are the correct dimensions of matrix A?

b. How many elements are in matrix A?

c. Suppose you wanted to multiply matrix A by another matrix B, so that you are calculating A · B. How many rows would matrix B need to have in order to do this calculation?

d. Suppose you decided to multiply matrix A by another matrix, C, which has dimensions of 3 rows and 8 columns. What would the dimensions be of this resulting matrix be by finding this product (A∗C)?

e. What would be the dimensions be of the resulting matrix by finding the product of (A*D), if D is a 2 x 3 matrix?

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#2. Short Answer: If A is a 4 x 2 matrix, B is a 3 x 7 matrix, and C is a 2 x 3 matrix, what are the dimensions of A x C x B?

Below are some problems that you can easily do without a calculator:

#3. Determine the sum [5 28 −5

1 −5 0 0

] + [2 2

−6 −6

2 2−3 −3

] if it exists.

#4. Determine the difference: [6 10 −97 −4 2

] − [8 9 40 4 −5

] if it exists.

#5. Determine what matrix is equivalent to X in the equation:

[6 812 −6

] + 𝑋 = [−8 012 6

]?

#6. Determine what matrix is equivalent to ‘D’ in the equation:

[4 1112 1014 12

] − 𝐷 = [0 00 00 0

]?

#7. Let W + Z = [0 0 00 0 0

00]. If W = [

−5 12 27 9 −6

], determine what matrix is equivalent to

Z?

#8. Determine what matrix is equivalent to −3 [10 6 −49 0 6

−7 −3 10]?

#9. Determine what matrix is the solution of:

[3 −2 614 0 −10

] − 2𝑋 = [3 4 56 2 6

] ?

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#10. The matrix [−4 −2 09 5 6

19 ] represents the vertices of a polygon. Determine what

matrix represents the vertices of the image of the polygon after a dilation of 4?

Below are some operations that you will need to be proficient in the use of the calculator for: #11. What is A∗B?

𝐴 = [

−3 16 04 29 7

] ; 𝐵 = [2 65 1

]

#12. Find the inverse of [4 6

−4 0 59]

#13. What is the inverse of this matrix? [2 6

−2 −4]

#14. Evaluate the determinant: |6 103 −8

|

#15. Evaluate the determinant: |5 6 70 0 83 −4 5

|

#16. What is the product of [7 −62 9

] ∗ [9

−54

]?

#17. What is the product of [123] [4 7 8]

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#18. Problem Solving: Four teams are going to participate in a speech competition. The number of 1st, 2nd, 3rd and 4th places finishes in each round determines the final score. The matrix shown shows the results of all 10 rounds of this competition. Teams will earn 4 points for each 1st place finish, 3 points for each 2nd place finish, 2 points for each 3rd finish, and 1 point for each 4th place finish. What is the ranking of the four teams?

NOTE: COLUMNS: The 1st column represents 1st place, the 2nd column represents the 2nd place, the 3rd column represents the 3rd place, and the 4th column represents the 4th place. ROWS: The 1st row represents Team 1, the 2nd row represents Team 2, the 3rd row represents Team 3, and the 4th row represents Team 4.

[

2 44 1

1 31 4

5 12 4

1 33 1

]

#19. Problem Solving: A used CD and DVD store sells DVDs for $5.00 each, CD’s for $3.00, and VHS tapes for $1.00 each. On a recent weekend, they sold 47 DVD’s, 38 CD’s, and 12 VHS tapes. Write a set of matrices that could show how you would compute the store’s total income for that day.

#20. CD/DVD Shop: At a cd and dvd bookstore, cd’s sell for $10 each and dvd’s sell for $15 each. You purchase 40 items and spend $450. How many cd’s did you buy?

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#21. Fruit Market: A fruit market is selling oranges in a 5 lb bag for $6 and a 10 lb bag for $10. You spend $68 and buy a total of 8 bags of oranges. Using a matrix, how many 5 lb bags and 10 lb bags of oranges did you buy? How many total pounds of oranges did you buy? #22. Numbers: The sum of three numbers is 21. The second number is two more than twice the first number. The second number is three times the third number. What are the numbers? #23. Geometry: One angle of a right triangle measures 90 degrees. The measure of the second angle is 5 times the measures of the third. What are the measures of these angles? #24. Thrift Shop: You walk into Goodwill with $20 in your pocket. The tags for t-shirts say that each shirt cost $1.00. The tags for the long-sleeve t-shirts say that each shirt cost $2.00. You spend exactly the amount in your pocket(s) (assume no sales tax) and walk out with 14 items of clothing. How many t-shirts did you buy? How many long-sleeve t-shirts did you buy?

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#25. What is the solution of the system?

{−𝟑𝒙 + 𝟐𝒚 = 𝟖𝒙 + 𝟐𝒚 = −𝟖

I can identify when a system of equations has NO SOLUTION or when it has INFINITELY MANY SOLUTIONS.

Consider the following:

#26. {3x – 2y = 8

4y = 6x – 5

#27. {2x + 8y = 6x = −4y + 3

#28. Write a matrix equation to express the following:

{

𝟒𝒙 + 𝟔𝒚 − 𝟏𝟎𝒛 = 𝟏𝟗−𝟓𝒙 + 𝟔𝒚 = 𝟐𝟕

𝒙 + 𝒚 − 𝟏𝟖𝒛 = 𝟐𝟎

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#29. What is the solution of the matrix equation below?

[𝟓 𝟑𝟑 𝟐

]𝑿 = [𝟏

−𝟑]

#30. What would be a system of equations that would represent the following matrix?

|−𝟓 𝟔 𝟒𝟎 𝟓 𝟏𝟕 𝟎 𝟎

|𝟕𝟏𝟖𝟑𝟕

||

#31. Consider this matrix: |10 8 70 −2 6

−3 5 0

9 52 816 −14

|

a. What is 𝑚24 in matrix M?

b. How many rows AND columns does the matrix have?

c. How many elements does the matrix have?

The frequency table shows the number of trees in the yard of each house on one street. #32. What are the mean, median, and mode for the trees per yard?

Trees 3 4 5 6 7 8

Yards 1 5 7 4 1 2

#33. Which number in the distribution shown below is an outlier? 56 65 73 59 98 65 59

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#34. The table shows population density by square miles for counties in three of Florida’s eight regions, according to the 2000 U.S. Census. Determine the mean, mode, range, five number summary, and interquartile range (IQR) for all of the counties regarding population density data. Additionally, after determining the quartiles of the population densities use a graphing calculator to create a boxplot.

Southwest Southeast Central East

204.2 228.1 401.9

13.7 573.0 467.7

548.6 1346.5 224.4

31.4 1157.9 46.4

124.1 79.8 336.6

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