Solving Matrices for Pmv & Garlic Bread Cost

[BilloRani2012]

Homework Statement

Phil bought 4 meat pizzas, 4 veg pizzas and one loaf of garlic bread, and it costs him $92. By Abyy, who is partial to garlic bread, bough 2 meat pizzas, 2 veg pizzas and 10 loaves of garlic bread, and it cost $84.

a) Use matrices to find Pmv, the most of 1 meat pizza and 1 veg pizza, and the cos of 1 loaf of garlic bread.

b) At Alex's party he bought 5 meat pizzas, 2 veg pizzas and 4 loaves of garlic bread. It cost him $96. Use matrices to calculate the cos of the different pizzas.

c) How much would 3 meat pizzas, 2 veg pizzas and 4 loaves of garlic breag cost?

Homework Equations

The Attempt at a Solution

For part a)
I let m= meat pizzas
v= veg pizzas
g= garlic bread

so my equations are: 4m+4v+1g = 92 and 2m+2v+10g = 84

so would my matrix look like this:
l 4 4 1 l = l m l
l 2 2 10 l l v l
l g l

 

[tiny-tim]

Hi BilloRani2012!

… so my equations are: 4m+4v+1g = 92 and 2m+2v+10g = 84

so would my matrix look like this:
l 4 4 1 l = l m l
l 2 2 10 l l v l
l g l

Not exactly …

| 4 4  1 |  | m | = |92|
| 2 2 10 |  | v |   |84|
            | g |

​

 

[uqphysicsguru]

i got this far... and then what? determinant can't be found because its not square

 

[tiny-tim]

hi uqphysicsguru!

to answer question a), i'd rewrite it as a 2x2 matrix with a 1x2 vector [m+v,g]

 

[BilloRani2012]

i'm a little confused...how would you write it as a 2x2 matix when we have 3 variables?

 

[BilloRani2012]

well since the first part of Q1a) says to find Pmv (Price of meat and veg pizza), would the 2x2 matix be this:

l 4 4 l l m l = l 91 l
l 2 2 l l v l l 74 l

would that be right? and just solve for m and v? BTW, i got 91 and 74 from my original equation. I subtracted the garlic bread numbers from the price.

so, my equations are: 4m+4v+1g = 92 and 2m+2v+10g = 84

92-1 = 91 and 84 - 10 = 74

i did this, becuase we are only trying to find Pmv for the the first part of Q1: a) not the garlic bread...

 

[gneill]

Give "1 meat pizza and 1 veg pizza" its own variable, say a. It turns out that Phil bought 4a and one loaf garlic bread, while Abyy bought 2a and ten loaves of garlic bread. There you go! Only two variables and two equations.

Note that the question did not ask for the individual price of meat pizza and veg pizza. It asked for the price of 1 meat pizza and 1 veg pizza together.

 

[BilloRani2012]

okay, so would my 2 equations be:

phill --> 4a + 1 = 92
abby --> 2a + 10 = 84

and convert that into a matix and solve for a?? because the question asks to solve using matrices!

 

[gneill]

okay, so would my 2 equations be:

phill --> 4a + 1 = 92
abby --> 2a + 10 = 84

and convert that into a matix and solve for a?? because the question asks to solve using matrices!

Sure. Looks like a good plan!

 

[BilloRani2012]

okay i did it using those2 equations and got
a = $22
g(garlic bread) = $4

and i substituted it back into the 2 equations:
phill --> 4a + 1 = 92
abby --> 2a + 10 = 84

and got 92 and 84...so is that right??

 

[gneill]

The numbers work, so they are correct.

Just be a bit careful about writing your equations. You wrote:

phill --> 4a + 1 = 92
abby --> 2a + 10 = 84

I understood your intention, but you should have included the second variable name (g I suppose, for garlic bread) in the formulation. Thus,

phill --> 4a + 1g = 92
abby --> 2a + 10g = 84

Just to make things neat and tidy.

 

[BilloRani2012]

oh yeppp...thanks :)

 

[BilloRani2012]

Can you please help me with part b) --> b) At Alex's party he bought 5 meat pizzas, 2 veg pizzas and 4 loaves of garlic bread. It cost him $96. Use matrices to calculate the cos of the different pizzas.

My Equation is: 5m + 2v + 4g = 96

but how would i calculate the price of each type of pizza?

 

[uqphysicsguru]

bread $4
veg $10
meat $12

 

[gneill]

Can you please help me with part b) --> b) At Alex's party he bought 5 meat pizzas, 2 veg pizzas and 4 loaves of garlic bread. It cost him $96. Use matrices to calculate the cos of the different pizzas.

My Equation is: 5m + 2v + 4g = 96

but how would i calculate the price of each type of pizza?

Remember how you had first laid out your matrix for part a, and you noticed that you had three variables but only two equations so that the matrix was not square?

Well, part b supplies another equation! Add this new information to that of part a and you can form the 3 x 3 matrix that you were looking for.

 

[Dazz4C]

bread $4
veg $10
meat $12

Don't know how you got that; but it should leave you with a matrix where the determinant is 0, thus the matrix does not have a unique answer and cannot be solved.

For example, although your answer for the prices is technically correct; I can say:
bread = $4
veg = $2
meat = $20

or

bread = $4
veg = $11
meat = $11

There is no definite answer using matrices.

EDIT: Nevermind, re read the second 'question' the OP posted. You should be correct.