# System of equations algebra question.

#### amathproblemthatneedsolve

##### New member

A band, The committee members of a local operatic and dramatic society organize a one-day music festival. It is held on two stages. The committee is trying to determine how many band acts (comprising two or more people) and how many solo acts (only one person) they will be able to hire for the day with an estimated $120 000 in total available to pay the acts. They want to have 30 acts (utilising the two stages) over the 24-hour period. I know this activity requires you to use systems of equation. And they want me to model the information provided about the fees paid to performers and to provide advice to the organizers on how many band acts and how many solo acts they should I think it;s also a good idea to relate the findings to the context. – b ±√b2 – 4ac 2a logarithms If y = bx then x = logby logb (xn) = n logbx If y = ex then x = logey (=1n y) Use Resource 1 to provide the organisers with the following information: 1. The change in the number of band acts in 2017 compared to 2016. 2. The fee they should offer to a band act in 2018 if: a. there are 30 acts in total b. all solo acts in will be offered a fee of$2 625 c. the total budget for acts is $120 000. Resource 1: resource 1 There are two different acts – a solo act and a band act. This is information from the previous two years (2016 and 2017) and the plan, so far, for 2018. 2016 • The number of acts was 22. • The relationship between the number of solo acts and band acts can be modelled by x2 + y = 402 where x is the number of solo acts and y represents the number of band acts. 2017 valley sounds • The amount to hire a solo act was$2 500 and the amount to hire a band act was $5 500 • The relationship between the number of solo acts and band acts can be modelled by xy = 50 + 3x where x is the number of solo acts and y represents the number of band acts. • The total cost of the acts was$90 000. 2018.

• The fee for a solo act is planned to be $2 625 and the fee for a band act has not yet been decided upon. • The relationship between the number of solo acts and band acts can be modelled by 133x + 400y2 = 59 994 where x is the number of solo acts and y represents the number of band acts. • The committee wants to have 30 acts. • The total budget for hire of acts is$120 000.

Where should I start all the words have slightly confused me??

#### Otis

##### Senior Member
Hello AMPTNS. Your post above is messy. It looks like maybe you tried copy-and-paste from online materials for some parts of your post but formatting was lost. You could have easily fixed all the mistakes, by editing. This is why we ask members to use the Preview button to proofread posts before submitting.

Disclaimer: The following comments (and paraphrased quotations) are based on some guesses.

… I know this activity requires … systems of [equations] … I think it's also a good idea to [make use of] the context [eg: quadratic formula and] logarithms …
I agree with using systems of equations, but I don't see any need for using logarithms. The equations are linear or quadratic, so each system may be solved by first using substitution to form a single equation in the standard form Ax^2+Bx+C=0 (or Ay^2+By+C=0, as needed) and then continuing by applying the quadratic formula.

… x is the number of solo acts and y represents the number of band acts …
Keep in mind that symbols x and y each represent a count of acts, so you must ignore any (x,y) solutions you get if either x or y is not a positive, Whole number.

(The sentence above has been edited for clarification.)

… provide … 1. The change in the number of band acts in 2017 compared to 2016
2016The number of acts was 22. The relationship between the number of solo acts and band acts can be modeled by x^2+y=402
2017 … The amount to hire a solo act was $2500 and … a band act was$5500. The relationship between the number of solo acts and band acts can be modeled by xy=50+3xtotal cost of the acts was $90000 That's enough information to write a system of equations for each year 2016 and 2017. Solve those systems, to find the number of band acts (y) in each year. Use those y-values to answer question 1. 2016 x + y = 22 x^2 + y = 402 2017 2500x + 5500y = 90000 xy = 50 + 3x … provide … 2. The fee they should offer to a band act in 2018 if … there are 30 acts in [total, each] solo [act receives]$2625 [and the] total budget … is $120000 … … The relationship between the number of solo acts and band acts can be modeled by 133x+400y^2=59994 We need to find the fee for band acts in 2018, so pick a symbol to represent that unknown. Let z = 2018 band fee Now we can model the total budget as: 2625x + zy = 120000 Next, we solve the following system (to find x and y). After that, substitute the values for x and y into the budget model above, and then solve for z. 2018 x + y = 30 133x + 400y^2 = 59994 If you'd like more help with this exercise, please post your work and any specific questions you have, and be sure to proofread your typing for clarity and completeness before submitting your posts. Thank you! … logey = 1n y PS: The name for the natural log function is ln, not 1n. (Weird typo?) Also, consider using function notation when working with functions: loge(y) = ln(y) Finally (in case you were unaware), we type a caret symbol ^ to show exponents. Like this: x^2 + y = 402 Last edited by a moderator: #### amathproblemthatneedsolve ##### New member Thanks, @Otis. Yes, I apologize for my messy post, thanks fur the useful information about the preview button! so for 2016 x + y = 22 x^2 + y = 402 (X1,Y1)= (20,2) for (X2,Y2)= (-19,41) So there was either 41 or 2 band acts in 2016? so for 2017 2500x + 5500y = 90000 (X1,X2)=(22/5,158/11) or (25,5) So there was either 158/11 or 5 band acts in 2017? xy = 50 + 3x for for 2018 x + y = 30 133x + 400y^2 = 59994 (X1,Y1) = (18,12) or (16667/400,4667,400) For all of my results I will use the positive and whole numbers as said by you " so you must ignore any solutions you get for x or y that are not positive, Whole numbers." Putting (18,12) into 2625x + zy = 120000 so Z=-2937.5. Not to sure if I've done any of this correct? #### Otis ##### Senior Member … For all of my results I will use the positive and whole numbers … Okay, but you don't seem to be doing that. … so for 2016 … (X1,Y1)=(20,2) … (X2,Y2)=(-19,41) So there was either 41 or 2 band acts in 2016? No. We need to ignore the solution (-19,41) because x=-19 is not positive. There were two band acts in 2016. … so for 2017 … (22/5,158/11) or (25,5) So there was either 158/11 or 5 band acts in 2017? … No. We need to ignore the solution (22/5,158/11) because x=22/5 and y=158/11 are not Whole numbers. There were five band acts in 2017. … for for 2018 … Putting (18,12) into 2625x + zy = 120000 so Z=-2937.5 … Note: The symbols z and Z are not the same (i.e., variable symbols are case-sensitive in math), so use symbols as given. z = -2937.5 cannot be correct because bands don't get paid with negative money. You must have made an arithmetic or substitution mistake somewhere. Additionally, if we note the given dollar amounts from 2016 through 2018, we see that fees increase over time. (That makes sense, yes?) Therefore, we ought to expect z to be greater than$5500.

Please double-check your calculations. If you're unable to find your mistake(s), then kindly show your steps (as requested in the Read Before Posting notice). This saves everybody time. Thank you.

By the way, are you member harpazo?

#### JeffM

##### Elite Member
Thanks, @Otis. Yes, I apologize for my messy post, thanks fur the useful information about the preview button!

so for
2016
x + y = 22
x^2 + y = 402 (X1,Y1)= (20,2) for (X2,Y2)= (-19,41) So there was either 41 or 2 band acts in 2016?

so for

2017
2500x + 5500y = 90000 (X1,X2)=(22/5,158/11) or (25,5) So there was either 158/11 or 5 band acts in 2017?
xy = 50 + 3x

for for
2018
x + y = 30
133x + 400y^2 = 59994 (X1,Y1) = (18,12) or (16667/400,4667,400) For all of my results I will use the positive and whole numbers as said by you " so you must ignore any solutions you get for x or y that are not positive, Whole numbers."

Putting (18,12) into 2625x + zy = 120000 so Z=-2937.5. Not to sure if I've done any of this correct?
$$\displaystyle x + y = 22 \implies y = 22 - x.$$

$$\displaystyle \therefore x^2 + y = 402 \implies x^2 - x + 22 = 402 \implies$$

$$\displaystyle x^2 - x - 380 = 0 \implies x = \dfrac{1 \pm \sqrt{1 + 4 * 380}}{2} = \dfrac{1 \pm \sqrt{1521}}{2} = \dfrac{1 \pm 39}{2} \implies$$

$$\displaystyle x = 20 \text { or } x = -\ 19.$$

But one of those answers is impossible? Which one and why?

#### amathproblemthatneedsolve

##### New member
@Otis So for
x + y = 30
133x + 400y^2 = 59994 this = (18,12) plugging this into 2625x + zy = 120000 I get . 2625(18) + z(12) = 120000 z= 6062.5?

Who on earth is harpazo?

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#### Otis

… z= 6062.5?
That's correct. The band act fee is $6,062.50 for 2018. Who on earth is harpazo? Some guy playing games at various math sites. He copies student posts and then pretends to be the student. (Using a different name, he claims to be a math teacher.) He had copied your post elsewhere (including the line "Where should I start all the words have slightly confused me?"). I was just curious to know whether amathproblemthatneedsolving was one of his many usernames. Please pay it no mind. #### amathproblemthatneedsolve ##### New member Ok, that's weird, anyway. Do I now have enough information to answer the questions? 1. The change in the number of band acts in 2017 compared to 2016. The change is there were 3 more acts in 2017 than 2016? 2. The fee they should offer to a band act in 2018 if The band act fee is$6,062.50 for 2018.

a. there are 30 total acts

b. all solo acts will be offered a fee of $2,656 c. the total budget for acts is$120,000

So I have all the answers, I believe ?

#### Otis

##### Senior Member
… 1. The change in the number of band acts in 2017 compared to 2016 … is there were 3 more acts in 2017 than 2016[.]

2. The fee they should offer to a band act in 2018 if The band act fee is \$6,062.50 for 2018.
Your answers are correct. (I would delete the text highlighted in red.)

The statements labeled a, b, and c are three pieces of given information which the author chose to label (i.e., those three lines are not part of the answer for question 2).