Question

A math test is made up of 5-point questions and 10-point questions. There are 24 questions on the test. The total number of possible points is 150 points. Write a system of linear equations to represent the problem. Use f for 5-point questions and t for 10-point questions. Find the intercepts of the graph of each equation you wrote in part a. State your answers in complete sentences. Use the intercepts from part b. to graph the system of linear equations. Label each axis to show what it represents. Label each line. What is the total value of all the 10-point questions on the test? Explain.

Answer 1

"Use f for 5-point questions and t for 10-point questions" means that
f = the number of 5-point questions in the test; and
t = the number of 10-point questions in the test

Now, express what we know:
The number of 5-point questions (f) plus (+) the number of 10-point questions (t) = (24) questions total
f + t = 24
The total points on the 5-point questions (5) times the number of 5-point questions (f) plus (+) the value of the 10-point questions (10) times the number of 10-point questions (t) = (150) total points possible
5f + 10t = 150

So, our system of linear equations is made up of two equations:
f + t = 24
5f + 10t = 150

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If you also need to solve:
Use the first equation to find the 'value' of f:
f + t = 24 isolate 'f' by making '+ t' equal to 0 (subtract 't' from both sides)
f + t - t = 24 - t
f + 0 = 24 - t
f = 24 - t

Now that we know that f = 24 - t, put in (24 - t) instead of 'f' in the second equation:
5f + 10t = 150
5(24 - t) + 10t = 150 Multiply both terms inside the brackets by 5:
5(24) + 5(- t) + 10t = 150
120 - 5t + 10t = 150 combine like terms (- 5t + 10t):
120 + 5t = 150 Now isolate 't' like we did for 'f' in the first equation:
120 + 5t - 120 = 150 - 120
0 + 5t = 30
5t = 30 Since '5t' means '5 multiplied by t', we need to divide everything by 5 to make the '5' equal to '1' and, thereby, '5t' equal to '1t', or, 't':
5t/5 = 30/5
1t/1 = 6
t = 6

Now that we know that t = 6, we go back to our first equation and put in '(6)' instead of 't', then solve for 'f' again:
f + t = 24
f + (6) = 24
f + 6 = 24
f + 6 - 6 = 24 - 6
f + 0 = 18
f = 18

Therefor, f = 18 and t = 6, which tells us that
there are 18 of the 5-point questions
and
there are 6 of the 10-point questions

Answer 2

\[f+t=24\]
\[5f+10t =150\]
\[t=24-f\]
\[5f+10(24-f)=150\]
\[5f+240-10f=150\]
\[-5f=-90\]
\[f=18\]
\[t-6\]

Answer 3

to graph these equations, you have to choose one of the variables to serve as "x" and the other as "y"

let t act as "x"
then f= -t+24

and in the second equation (after dividing everything by 5)
f= -2t+30

Answer 4

Hello BrizyBaby16, I am Trackstar84. I am also a FLVS student. I have been working on this problem but could not solve it. I got as far as turning the word problem into an equation. since your version was so clear and concise, would you mind sharing with me how you were able to brake the problem apart and solve it? After reading the word problem and putting it into an equation, how did you know what to do from there?

Answer 5

@TrackStar84 what equations did you get?

Answer 6

Thank you PHI, with your help and the example above, I was able to solve my issue. Sorry I did not get back to you sooner.