Exam 10: Systems and Matrices

Perform the operation or operations when possible. - $\left[ \begin{array} { r } 9 x + 2 y \\3 k - 7 z \\- 6 w + 9 v \\8 m - 9 n\end{array} \right] - \left[ \begin{array} { r } - 2 x - 5 y \\- 7 k - 3 z \\- 4 w + 5 v \\4 m - 5 n\end{array} \right]$

Provide an appropriate response. -Suppose that you are solving a system of three linear equations by the Gauss-Jordan method and obtain the following augmented matrix. $\left[ \begin{array} { r r r | r } 1 & - 10 & - 5 & 8 \\0 & 1 & 9 & 16 \\0 & - 8 & 5 & 6\end{array} \right]$ What row transformation would you perform next?

A nonlinear system is given, along with the graphs of both equations in the system. Determine if the points of intersection specified on the graph are solutions of the system by substituting directly into both equations. - +=25 2y+x=5

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. -A chair manufacturing company has two different factories. The profit function for Factory I when $x$ are manufactured and sold is as follows: $P ( x ) = - 0.05 x ^ { 2 } + 20 x - 500$ The profit function for Factory II when $x$ chairs are manufactured and sold is as follows: $P ( x ) = - 0.05 x ^ { 2 } + 30 x - 2500$ For what number of chairs is the profit the same at both factories, and what is the profit?

Find the matrix product when possible. -Le Boulangerie, a bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of e. 44 ingredient (in cups, except for eggs) required for these items is given by matrix A. Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders.

Solve the linear programming problem. -An airline with two types of airplanes, $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ , has contracted with a tour group to provide transportation for a minimum of 400 first class, 900 tourist class, and 1500 economy class passengers. For a certain trip, airplane $P _ { 1 }$ costs $\$ 10,000$ to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane $P _ { 2 }$ costs $\$ 8500$ to operate and can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost?

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. -A bookstore is having a sale. All books included in the sale have a colored sticker on them to indicate the sale price. There are green stickers, red stickers, and orange stickers. Bob, Sue, and Fred each make purchases of books that are on sale. Each row of the table gives information about the numbers of book purchases and the total cost of the purchase (before taxes). Person Green Red Orange Total Cost Bob 1 2 2 \ 29.04 Sue 1 3 2 \ 35.91 Fred 1 2 3 \ 34.21 Use this information to set up a matrix equation of the form $\mathrm { AX } = \mathrm { B }$ , which can be solved to determ: the price for each type of sale book. Solve this matrix equation to find the price of a book with a red : Use the fact that for $A = \left[ \begin{array} { l l l } 1 & 2 & 2 \\ 1 & 3 & 2 \\ 1 & 2 & 3 \end{array} \right] , A ^ { - 1 } = \left[ \begin{array} { r r r } 5 & - 2 & - 2 \\ - 1 & 1 & 0 \\ - 1 & 0 & 1 \end{array} \right]$ .

Graph the solution set of the system of inequalities. - y\geq y\leq8

Solve the problem. -A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. $\mathrm { C }$ requires 2 black, 1 white, and 2 red. The company used 100 black, 110 white and 80 red wires. How many of each type of cable were made?

Write the augmented matrix for the system. Do not - 5x+5y+6z=61 7x+4y+3z=74 8x+7y-2z=84

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $y \leq - 3$

If the system has infinitely many solutions, write the solution set with x arbitrary. - 10x-y=5 -20x+2y=-10

Find the value of the determinant. - $\left| \begin{array} { l l } 1 & - 6 \\8 & - 9\end{array} \right|$

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. -Which of the following properties do not hold for matrix multiplication? (i) Associative property (ii) Commutative property (iii) Distributive property (iv) Zero-factor property (If the product of two matrices is a zero matrix, then one of the factors must be a zero matrix.)

Graph the solution set of the system of inequalities. - x-2y\leq2 x+y\leq0

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - x-y=7 100x+y=9

Find the matrix product when possible. -Momma's ice cream shop sells three types of ice cream: soft-serve, chunky, and nonfat. Location I sells 14 gal of soft-serve, 80 gal of chunky, and $30 \mathrm { gal }$ of nonfat ice cream each day. Location II sells 28 gal of soft-serve and Location III sells 60 gal of soft-serve each day. Daily sales of chunky ice cream are 90 gal at Location II and 120 gal at Location III. At Location II, 38 gal of nonfat are sold each day, and $40 \mathrm { gal }$ of nonfat are sold each day at Location III. Write a $3 \times 3$ matrix that shows the sales figures for the three locations, with the rows representing the three locations. The incomes per gallon for soft-serve, chunky, and nonfat ice cream are $\$ 7 , \$ 3$ , and $\$ 6$ , respectively. Write a $3 \times 1$ matrix displaying the incomes. Find a matrix product that gives the daily income at each of the three locations.

Write the system of equations associated with the augmented matrix. Do not solve. - $\left[ \begin{array} { r r r | r } 5 & 6 & - 9 & - 9 \\6 & 5 & 0 & - 4 \\- 3 & 0 & 3 & 7\end{array} \right]$

Write the system of equations associated with the augmented matrix. Do not solve. - $\left[\begin{array}{rr|r}11 & 18 & 3 \\3 & 2 & 15\end{array}\right]$

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - 3x-2y-4z+1=0 9x-2y+8z-4=0 4y+4z-3=0