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Five-Minute Check (over Lesson 3–4) Then/Now New Vocabulary Example 1: Real-World Example: Analyze Data with Matrices Key Concept: Adding and Subtracting Matrices Example 2: Add and Subtract Matrices Key Concept: Multiplying by a Scalar Example 3: Multiply a Matrix by a Scalar Key Concept: Properties of Matrix Operations Example 4: Multi-Step Operations Example 5: Real-World Example: Use Multi-Step Operations with Matrices Lesson Menu

maximum: f(4, 4) = 4 minimum: f(4, 11) = –10 Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum an minimum values of the given function for this region. 1 ≤ x ≤ 4, y ≥ x, y ≥ 2x + 3; f(x, y) = 3x – 2y maximum: f(4, 4) = 4 minimum: f(4, 11) = –10 B. maximum: f(4, 11) = 10 minimum: f(1, 1) = 1 C. maximum: f(4, 4) = 4 minimum: f(1, 5) = –7 D. maximum: f(4, 11) = 10 minimum: f(4, 4) = 4 5-Minute Check 1

A company profits $3 for every widget it manufactures and $2 for every plinket it manufactures. It must make at least one widget and one plinket each hour, but cannot make more than 7 total widgets and plinkets in any hour. What is the most profit the company can make in any hour? A. $21 B. $20 C. $18 D. $5 5-Minute Check 2

You organized data into matrices. Analyze data in matrices. Perform algebraic operations with matrices. Then/Now

scalar multiplication Vocabulary

Analyze Data with Matrices A. Use the matrix below that includes information on tuition (T), room and board (R/B), and enrollment (E) for three universities. ISU UI UNI T R/B E Find the average of the elements in column 1, and interpret the result. Example 1

Answer: The average tuition cost for the three universities is $5935. Analyze Data with Matrices Answer: The average tuition cost for the three universities is $5935. Example 1

Which university’s total cost is the lowest? Analyze Data with Matrices B. Use the matrix below that includes information on tuition (T), room and board (R/B), and enrollment (E) for three universities. ISU UI UNI T R/B E Which university’s total cost is the lowest? Example 1

Answer: University of Northern Iowa Analyze Data with Matrices ISU = 6160 + 5958 = $12,118 UI = 6293 + 7250 = $13,543 UNI = 5352 + 6280 = $11,632 Answer: University of Northern Iowa Example 1

Analyze Data with Matrices C. Use the matrix below that includes information on tuition (T), room and board (R/B), and enrollment (E) for three universities. ISU UI UNI T R/B E Would adding the elements of the rows provide meaningful data? Explain. Answer: No, the first two elements of a row are in dollars and the third is in numbers of people. Example 1

Analyze Data with Matrices D. Use the matrix below that includes information on tuition (T), room and board (R/B), and enrollment (E) for three universities. ISU UI UNI T R/B E Would adding the elements of the third column provide meaningful data? Explain. Answer: Yes, the sum of the elements of the third column would be the total enrollment of all three schools. Example 1

Execution Degree Score of Difficulty The matrix displays Karen’s diving scores for her 6 dives at a competition. The total score is found by multiplying the degree of difficulty by the execution score. Dive 1 Dive 2 Dive 3 Dive 4 Dive 5 Dive 6 Execution Degree Score of Difficulty Example 1

A. Find the average of the elements in column 1, and interpret the results. A. The average number of dives is 8.36. B. The average score for the 6 dives is 8.36. C. The average execution for the 6 dives is 8.36. D. The average degree of difficulty for the 6 dives is 8.36. _ Example 1

B. The matrix displays Karen’s diving scores for her 6 dives at a competition. The total score is found by multiplying the degree of difficulty by the execution score. Which dive’s total is the highest? A. dive 1 B. dive 3 C. dive 4 D. dive 6 Example 1

A. Yes, adding the elements gives the total score. C. The matrix displays Karen’s diving scores for her 6 dives at a competition. The total score is found by multiplying the degree of difficulty by the execution score. Would adding the elements of the rows provide meaningful data? Explain. A. Yes, adding the elements gives the total score. B. No, the last element of the row is the product of the first and second elements in the row. Example 1

D. The matrix displays Karen’s diving scores for her 6 dives at a competition. The total score is found by multiplying the degree of difficulty by the execution score. Would finding the average of the last column provide meaningful data? A. Yes, the average of the last column would be the average score for all 6 dives in the competition. B. No, each score has a different degree of difficulty, so you can’t find the average. Example 1

Concept

Add corresponding elements. Add and Subtract Matrices Substitution Add corresponding elements. Simplify. Answer: Example 2

Add and Subtract Matrices – Answer: Since the dimensions of A are 2 × 3 and the dimensions of B are 2 × 2, these matrices cannot be subtracted. Example 2

A. B. C. D. Example 2

A. B. C. D. Example 2

Concept

Multiply a Matrix by a Scalar Substitution Example 3

Multiply each element by 2. Multiply a Matrix by a Scalar Multiply each element by 2. Simplify. Answer: Example 3

A. B. C. D. Example 3

Concept

Perform the scalar multiplication first. Then subtract the matrices. Multi-Step Operations Perform the scalar multiplication first. Then subtract the matrices. 4A – 3B Substitution Distribute the scalars in each matrix. Example 4

Subtract corresponding elements. Multi-Step Operations Multiply. Subtract corresponding elements. Simplify. Answer: Example 4

A. B. C. D. Example 4

Use Multi-Step Operations with Matrices BUSINESS A small company makes unfinished desks and cabinets. Each item requires different amounts of hardware as shown in the matrices. DESK Short Long Nails Screws CABINET The company has orders for 3 desks and 4 cabinets. Express the company’s total needs for hardware in a single matrix. Example 5

Write matrices. Multiply scalars. Add matrices. Short Long Nails Use Multi-Step Operations with Matrices Write matrices. Multiply scalars. Add matrices. Short Long Nails Screws Answer: Example 5

Blue Yellow Green A. B. C. D. Course A Course B Course C Miniature golf course A has 50 blue golf balls, 100 yellow golf balls, and 50 green golf balls. Miniature golf course B has 150 blue golf balls, 100 yellow golf balls, and 25 green golf balls. Miniature golf course C has 40 blue golf balls, 70 yellow golf balls, and 80 green golf balls. Express the total number of each color golf ball in a single matrix. Blue Yellow Green A. B. C. D. Course A Course B Course C Example 5

End of the Lesson