Question

PLEASE HELP WILL FAN AND MEDAL
A bakery sells muffins for $3.50 each. A beverage is $1.75. A class purchases 32 items and spends a total of $87.50.
A. Define your variables. Write the system of equations and represent it as a matrix equation.
B. State the value of the determinant.
C. Use the matrices to solve the system. Find the number of muffins and the number of beverages purchased.

Show ALL work.

Answer 1

https://mathway.com/

Answer 2

I can't because mathway only gives answers

Answer 3

So how would you set up your equations?

Answer 4

@AlexandervonHumboldt2

Answer 5

You don't want me to help?

Answer 6

@Here_to_Help15

Answer 7

@confluxepic

Answer 8

@whpalmer4

Answer 9

Hmm, I never solve these as matrices, let me think :-)

Okay, what are the two equations that represent this scenario?

Answer 10

You have two different items being bought/sold, right?

Answer 11

yes,

Answer 12

okay, pick a variable name for each item being sold

Answer 13

now, can you write the equation that describes how many of each item is sold to the total number of items sold?

Answer 14

I don't have any time left, could we move a bit faster?

Answer 15

I'm bad at that.

Answer 16

I waited several minutes for you to respond, then moved on when you did not do so.

Answer 17

I lost battery and was looking for my charger

Answer 18

Okay, well, we can try to work it now, if you're all plugged in...

Answer 19

Yes I'm plugged in now. Sorry!

Answer 20

Okay, so I'm going to repost the salient bits here so I don't have to scroll back:

A bakery sells muffins for $3.50 each. A beverage is $1.75. A class purchases 32 items and spends a total of $87.50.
A. Define your variables. Write the system of equations and represent it as a matrix equation.

Answer 21

what is your variable for the number of muffins going to be called? how about beverages?

Answer 22

x for muffins, y for beverages

Answer 23

okay. not too descriptive, but they'll work :-)

so, can you write an equation that says that the class purchases 32 muffins and beverages?

Answer 24

or here how about m and b so its easier

Answer 25

ummmm, I'll try.

Answer 26

3.50m+1.75b=32 ??

Answer 27

no, we don't care about prices here...just the number of items

Answer 28

3.50 and 1.75 are the prices of those items, right?

Answer 29

Oh so m+b=32

Answer 30

yes

Answer 31

we buy 32 items, and the only items we buy are beverages and muffins so that is correct

Answer 32

that's the easier of the two to write. Now we need to write one that expresses how much money we spent. this one will have the prices, but on the other side of the equals sign it will have the amount of money we spent in total

Answer 33

3.50m+1.75b=87.50

Answer 34

yes, very good. so we have the following two equations describing our scenario:

\[m + b = 32\]\[3.50m + 1.75b = 87.50\]

Answer 35

Now we have to write that as a matrix equation

Answer 36

I'm not good at that... help?

Answer 37

[ 1 1 : 32 ]
[3.5 1.75 : 87.5] is that right?

Answer 38

well, the first step is to write the equations in the same variable order, as i did


\[\left[\begin{matrix}1 & 1\\ 3.5 & 1.75\end{matrix}\right]\left(\begin{matrix}m \\ b\end{matrix}\right)= \left(\begin{matrix}32 \\ 87.50\end{matrix}\right)\]

Answer 39

I didn't typeset that quite right, I guess.

Answer 40

I can see it fine

Answer 41

what's next?

Answer 42

Now we need to find the determinant of that coefficient matrix, I believe

Answer 43

How do I do that?

Answer 44

well, the determinant of a 2x2 matrix \[A =\left[\begin{matrix}a & b\\ c & d\end{matrix}\right] \] is given by \[\left| A \right| = ad - bc\]

Answer 45

1.75?

Answer 46

missing the leading negative sign, aren't you?

Answer 47

-1.75? haha

Answer 48

Yes, that's better. Now, have you learned how to use matrices to solve equations like this?

Answer 49

wait no I don't think i am 3.5-1.75=1.75

Answer 50

our matrix is

[1 1]
[3.5 1.75]

right?

[a b]
[c d]

determinant is \(ad-bc\) so that gives us \[1*1.75 - 1*3.5 = 1.75-3.5 = -1.75\]

Answer 51

oh. I did it wrong then.

Answer 52

is the answer m=18 b=14

Answer 53

@whpalmer4

Answer 54

okay, do you agree with my result?

Answer 55

if so, we've done A and B. to do C, we need to solve the matrix. there are a number of different ways to do that, and we should go with a way you've been taught, but I don't know which way(s) you've been taught...