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tudents are given 3 minutes for each multiple-choice question and 5 minutes for each free-response question on a test. There are 15 questions on the test, and students are given 51 minutes to take it. The system of equations shown can be used to find the number of multiple-choice questions, m, and the number of free-response questions, f, on the test. m + f = 15 3m + 5f = 51 How many multiple-choice questions are on the test? 3 5 12 14

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mbh292
In this question, they gave you the system of equations. You just need to use the two of them.

We want to find the m or the number of the multiple-choice questions. So we need to turn our last equation in the form of 'm's by translating the 'f's into 'm's.

[tex]m + f = 15[/tex]
[tex]f = 15 - m[/tex]

If we know the m form of f, we can convert the second equation and find the answer: [tex]3m + 5f = 51[/tex]

Plug in the m value of f: [tex]3m + 5(15 - m) = 51[/tex]

Distribute the 5: [tex]3m + 75 - 5m = 51[/tex]

Subtract -75 from both sides: [tex] 3m - 5m= 51 - 75[/tex]

Combine like terms: [tex] - 2m = - 24[/tex]

Divide both sides by -2: [tex]m = 12[/tex]

So, we learned that the number of the multiple-choice questions is 12.

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