Algebra Problems of the Day (Algebra Regents, August 2024 Part IV)

This exam was adminstered in August 2024 .

August 2024 Algebra, Part IV

Each correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. Jen joined the Fan Favorite Movie Club at the local movie theater. At this theater, the cost of admission in May and June remains the same. In May, she saw 2 matinees and 3 regular-priced shows and spent $38.50. In June, she went to 6 matinees and one regular-priced show and spent $47.50.

Write a system of equations to represent the cost, m, of a matinee ticket and the cost, r, of a regular-priced ticket.

Jen said she spent $5.75 on each matinee and $9 on each regular show. Is Jen correct? Justify your answer.

Use your system of equations to algebraically determine both the actual cost of each matinee ticket and the actual cost of each regular ticket.

Answer:

You need to write a system of equations that models the prices of the movie tickets. Substitute her numbers into the equations to see if they work. (They probably won't because of the next part of the problem.) Then solve that system using elimation or substituion.

The system of equations that models the ticket buying is as follows:

2m + 3r = 38.50
6m + 1r = 47.50

Jen believes that m = 5.75 and r = 9, so plug those values in:

2(5.75) + 3(9) = 38.50
6(5.75) + 1(9) = 47.50

38.50 = 38.50 (check)
43.50 =/= 47.50 (does not check)

Jen is incorrect because her values don't work for the month of June.

You can use substitution with the second equation, or you could multiply either equation by 3 and subtract.

2m + 3r = 38.50
6m + 1r = 47.50

6m + 9r = 115.50
6m + 1r = 47.50
8r = 68.00
r = 8.50

2m + 3(8.50) = 38.50
2m + 25.50 = 38.50
2m = 13.00
m = 6.50

Check: 6(6.50) + 8.50 = 47.50 (Check)

A matinee ticket is $6.50 and a regular ticket is $8.50.

Using substitution:

2m + 3r = 38.50
6m + 1r = 47.50
r = -6m + 47.50

2m + 3(-6m + 47.50) = 38.50

2m - 18m + 142.50 = 38.50
-16m = -104.00
m = 6.50

2(6.50) + 3r = 38.50
13 + 3r = 38.50
3r = 25.50
r = 8.50

End of Part Exam

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