Sales Commissions A saleswoman works at a kiosk that offers...

Sales Commissions A saleswoman works at a kiosk that offers three different models of cell phones: standard with 16 GB capacity, deluxe with 32 GB capacity, and super-deluxe with 64 GB capacity. For each phone that she sells, she earns a commission based on the cell phone model. One week she sells 9 standard, 11 deluxe, and 8 super-deluxe and makes $\$ 740$ in commission. The next week she sells 13 standard, 15 deluxe, and 16 super-deluxe for a $\$ 1204$ commission. The third week she sells 8 standard, 7 deluxe, and 14 super-deluxe, earning $\$ 828$ in commission. (a) Let $x, y,$ and $z$ represent the commission she earns on standard, deluxe, and super-deluxe, respectively. Translate the given information into a system of equations in $x, y,$ and $z$ (b) Express the system of equations you found in part (a) as a matrix equation of the form $A X=B$ . (c) Find the inverse of the coefficient matrix $A$ and use it to solve the matrix equation in part (b). How much commission does the saleswoman earn on each model of cell phone?

Sales Commissions A saleswoman works at a kiosk that
offers three different models of cell phones: standard with
16 GB capacity, deluxe with 32 GB capacity, and super-deluxe with 64 GB capacity. For each phone that she sells,
she earns a commission based on the cell phone model. One
week she sells 9 standard, 11 deluxe, and 8 super-deluxe and
makes $\$ 740$ in commission. The next week she sells 13 standard, 15 deluxe, and 16 super-deluxe for a $\$ 1204$ commission. The third week she sells 8 standard, 7 deluxe, and
14 super-deluxe, earning $\$ 828$ in commission.
(a) Let $x, y,$ and $z$ represent the commission she earns on standard, deluxe, and super-deluxe, respectively. Translate the
given information into a system of equations in $x, y,$ and $z$
(b) Express the system of equations you found in part (a) as
a matrix equation of the form $A X=B$ .
(c) Find the inverse of the coefficient matrix $A$ and use it to
solve the matrix equation in part (b). How much commission does the saleswoman earn on each model of cell
phone?

Key Concepts

Translating Real-World Problems into Systems of Equations

This concept involves taking a real-world scenario—in this case, sales commissions—and expressing the relationships algebraically by assigning variables to unknown quantities and forming equations that represent the given conditions. It includes understanding how each piece of provided numerical data (such as numbers of items sold and total commission earned) contributes to forming the equations that model the situation.

Matrix Formulation of Linear Equations

This key concept converts a set of linear equations into a matrix equation of the form A X = B, where A is the coefficient matrix, X is the vector of variables, and B is the vector containing the constants from the equations. This formulation offers a compact representation of the system and is a critical step for applying linear algebra methods.

Inverse Matrix Method in Solving Systems

The inverse matrix method is a technique used to solve a linear system represented as A X = B. When the coefficient matrix A is invertible, the solution can be found by computing X = A?¹B. This involves finding the inverse of A—which is itself a key linear algebra operation—and then performing matrix multiplication with the constant vector B to yield the solution for the variables.

Transcript

00:01 In this problem, we are given a lot of information about a saleswoman and the different amounts of cell phones that she sells and the commission that she makes for each week.

00:13 So part of a problem asks us to let x, y, and z represent the commission on standard deluxe and super deluxe cell phones.

00:22 And we want to write this information as a system of equations.

00:27 So we know that the standard is going to correspond to x, deluxe model will correspond to y and super deluxe will correspond to z so essentially each of our weeks is going to correspond to one of our equations in our system so for week one we know that we have nine standards so nine x plus 11 times our deluxe which will make 11 y plus eight times our super deluxe and this is going to give us seven hundred and forty dollars on week two we're going to sell 12 standard models so 12 x plus 15 deluxe y plus 16 super deluxe z, which is going to give us $1 ,204.

01:08 And finally, on our third week, we have three standard models, so 3x, plus 7 deluxe, 7y, plus 14 super deluxe z, is going to be equal to $828.

01:23 So that's part a of our problem.

01:25 For part b of our problem, we want to write this system of equations as a matrix equation.

01:31 So essentially we're just going to be able to write this.

01:36 I actually do that on the top up here as a coefficient matrix, which we know is going to be in our first equation.

01:45 We have 9, 11, 8.

01:48 Our second equation has coefficients of 12, 15, and 16.

01:53 And our third equation has coefficients 3, 7, and 14.

01:59 And we're multiplying this by our variable matrix.

02:03 Which is x, y, z, and this will be equal to the constants on the right -hand side of our equation, so equal to 740, 1 ,204, and 828.

02:18 And finally, for part c of our problem, we want to find the inverse of the coefficient matrix a, and use it to solve the matrix equation in part b.

02:28 And so once we do that, it's going to answer the question, how much commission does the saleswoman earn on each model of cell phone? so we can go ahead and start working on finding our inverse of our matrix here.

02:45 I'll go ahead and give us some extra space to work.

02:50 As we know that we're going to write this out as a 3x6 matrix, where the left 3x3 matrix is going to be our original matrix that we're given, 3 ,714.

03:06 And then on the right -hand side we're going to draw a dash line and then write our identity matrix out and so essentially all we're going to do is row reduce so that the left -hand side of our equation is going to give us a new identity matrix so right away we can see that regardless of what we do this is going to give us quite a few fractions here so i'm going to go ahead and begin by dividing my entire first row so, actually, i'll go ahead and start with our third row.

03:44 We can go ahead and compute row two minus four times our third row, and row one minus three times our third row.

03:56 And so essentially the reason that i chose to do this was because i know it's going to give us some zeros in our positions there.

04:04 So our top row, row one minus three, row three.

04:08 We get nine minus nine, which is zero.

04:11 11 minus 3 times 7 so 11 minus 21 will give us negative 10 and then we have 8 minus 14 times 3 and so when we multiply that out and subtract that's going to leave us with negative 34 and then we have 1 0 negative 3 our second row we have row 2 minus 4 times row 3 so we have 12 minus 12 which is 0 of 15 minus 4 times 7 so 15 minus 2 is 7 so 15 minus 28 is going to to be negative 13.

04:44 And then we have 16 minus 4 times 14, which is going to give us negative 40.

04:51 And then we have 0 1, negative 3.

04:54 And our third row is going to stay the same.

04:56 3 ,7, 14, 0 ,0.

05:04 And from here we can see that we now have nothing that's going to cancel really nicely.

05:10 So i'm going to go ahead and start by dividing my top equation by 10, or my top row by 10.

05:22 So row 1 over 10...

Please give Ace some feedback