Help with matrix story problem

Question

 

A hardware store sells three types of shovels, spade, flat, and square. The spade costs $16.00, the flat costs $9.60, and the square costs $12.80 each. The store keeps an inventory of 80 shovels. They always buy twice as many spade shovels as square. One day, after selling out of shovels, the store had receipts totaling $1,072.00. Find how many of each type of shovel were sold that day.

Write a system of equations to represent the information given.

Put the system into a matrix. Name the matrix using dimensions.

What is the determinant of the coefficients?

Using Cramer’s rule, how would you solve the system?

If the store doubled all prices and inventory, what would the new matrix be?

I'm just very confused on how to set-up this question, and how to apply it to the following questions. Any help is appreciated :)

Answer 1

I can't post a video right now due to maintenance, so I'll do my best to type this all out in a way that makes sense.

We are told there are 3 types of shovels, and I will define them as x,y,and z

x=spade

y=flat

z=square

We are also told that the store keeps 80 total shovels in their inventory. We can write our first eq. using this info: x + y + z = 80

We are also given the prices of each kind and the total money the store makes if they sell their whole inventory. This can be used to write another equation: 16x + 9.6y + 12.8z = 1072

Since we have three unknown variables, we need three equations. The problem tells us that they always carry twice as many spades as they do square shovels. This allows for another equation : 2z = x

the equations must all follow the same form to be out into a matrix, so we will rewrite this one as x + 0y - 2z = 0

we can write this now in matrix form [ 1 1 1 [ x [ 80

16 9.6 12.8 * y = 1270

1 0 -2 ] z ] 0 ]

The first 3X3 matrix is the coefficient matrix followed by the variables, followed by the solutions

*if the store doubled their prices and their inventory, the price equation would be 2*(16x + 9.6y + 12.8z) = (1072*2) then you can simplify and update the coefficients in your matrices!

Hope this answers at least some of your questions!

Answer 2

x = # of spades

y = # of flats

z = # of squares

"The store keeps an inventory of 80 shovels. They always buy twice as many spade shovels as square."

This means there must be some sort of combination of all the number of spades, flats, squares that adds up to 80 shovels AND when there is 2x the amount of squares then spades.

Equation of total # of shovels:

x + y + z = 80 (total number of shovels)

2z = x (2 times the amount of spades gives the number of squares)

As there is 3 variables, 3 equations is needed to find the value of these variables.

"One day, after selling out of shovels, the store had receipts totaling $1,072.00."

Need to combine the prices of these shovels to reach 1,072.

16x = price of spades

9.6y = price of flats

12.8z = price of squares

16x + 9.6y + 12.8z = 1072

System of Equations:

x + y + z = 80

2z = x (-x + 0y + 2z = 0)

16x + 9.6y + 12.8z = 1072

In Matrix Form

x y z

| 1 1 1 |

| -1 0 2 |

| 16 9.6 12.8|

3x3 Matrix

Determinant of 3x3 Matrix:

Editor's Note: I personally like to expand out the matrix and do it by 2x2 matrices as I do it as such for cross products.

D = 1(0*12.8 - 2 * 9.6) - 1(-1*12.8 - 2*16) + 1(-1*9.6 - 0 *16)

D = (-19.2) - (-44.8) + (-9.6)

D = 16

Dx: (substitute answers in x column)

x y z

| 80 1 1 |

| 0 0 2 |

| 1072 9.6 12.8|

Dx = 80(0*12.8 - 2*9.6) - 1(0*12.8 - 2*1072) + 1 (0 *19.6 + 0*1072)

Dx = 80(-19.2) - (2144) + 0

Dx = 608

Dy:

x y z

| 1 80 1 |

| -1 0 2 |

| 16 1072 12.8|

Dy = 1(0*12.8 - 2*1072) - 80(-1*12.8-2*16) + 1(-1*1072-0*16)

Dy = -2144 - 80(-44.8) - 1072

Dy = 368

Dz:

x y z

| 1 1 80 |

| -1 0 0 |

| 16 9.6 1072|

Dz = 1(0*1072 - 0*9.6) - 1(-1*1072-0*16) + 80(-1*9.6-0*16)

Dz = 0 - (-1072) + 80(-9.6)

Dz = 304

Final Solution:

x = Dx/D

x = 608/16 = 38

y = Dy/D

y = 368/16 = 23

z = Dz/D

z = 304/16 = 19

Answer 3

It's been awhile since I've done matrices to be honest. You don't need one to solve this though, just some algebra. Honestly, it's whatever is more comfortable to you, though. If you are comfortable with matrices, that's even better! It's just been a few years since I've had to use them. I could probably solve it much faster just using algebra, honestly. Knowing how to do it both ways could help you to understand it much better as a whole.

So I use elimination and substitution to solve the equations. Doing it this way allows you to work with just two equations. When you have more than two, its easier to get confused. So, I just adjust at the beginning to make sure I only have two variables and two equations.

Let A represent spade, Let B represent flat, Let C represent square

If they keep an inventory of 80 shovels, then A + B + C = 80. We are also told that they have twice as many spade shovels as they do square shovels. so we can adjust this equation to reflect that.

A + B + (1/2)A = 80 or

1.5A + B =80

The spades cost $16, the flats cost $9.60, and the squares cost $12.80. The price of their inventory totals to $1,072. So the equation is 16A + 9.6B + 12.80C = 1,072. Now we can adjust this equation as well so that we only have two variables.

16A + 9.6B + 12.80(1/2A) = 1,072

16A + 9.6B + 6.4A = 1,072

22.4A + 9.6B = 1,072

Now we can put the two equations together. Next, we need to manipulate at least one of the equations so that one of the variables disappears, leaving us with only one variable to solve for.

22.4A + 9.6B = 1,072

1.5A + B =80

If I multiply the bottom equation by (-9.6), we will have 9.6B and -9.6B, and they will cancel.

(1.5A + B =80) X (-9.6)

-14.4A - 9.6B = -768

Now we can add them and solve.

-14.4A - 9.6B = -768

+ 22.4A + 9.6B = 1,072

8A = 304

A = 304/8 = 38

So, there are 38 spade shovels. Since there are half as many square shovels as there are spades, then there are 19 square shovels. The remaining 23 shovels are flat.

Answer 4

1) Define your variables:

x - spade

y - flat

z - square

2) Set-up a system of equations:

x+y+z = 80 (equation for total amount of items sold)

16x+9.6y+12.8z = 1,072 (equation for total amount of money made )

x = 2z (the problem mentions that the amount of spades sold is twice the amount of squares sold)

3) Plug in 2z for x in the equations for profit made and how many items were sold:

3z+y = 80

44.8z+9.6y=1,072

(This is your system of equations)

4) Set up the 2x2 matrix:

(Note the matrices are slightly shifted due to the Wyzant platform, so make sure to line up the values, ie 2x3)

⌈a₁ b₁ | c₁ ⌉ => ⌈3 1 | 80 ⌉

⌊a₂ b₂ | c₂ ⌋ ⌊44.8 9.6 | 1072 ⌋

5) Solve using Cramer's Rule

z = Dz/D, y=D_y/D where D is the Determinant and D_x & D_y are the determinants for the x and y matrices

(Note in examples of Cramer's rule x is usually solved for instead of z so you would normally see x = D_x/d)

D = a₁×b₂ - b₁×a₂

= 3(9.6) - 1(44.8)

= -16

Dz = c₁×b₂ - b₁c₂

= 80(9.6) - 1(1072)

= -304

Dy = a₁×c₂ - c₁×a₂

=3(1072) - 80(44.8)

= -368

z = D_z/D

= -304/-16

= 19

y = D_y/D

= -368/-16

= 23

x = 2z

x = 2(19)

= 38

6) If the store doubled its prices the matrix would look like this:

⌈ 3 1 | 160 ⌉

⌊ 89.6 19.2 | 2144⌋