Exam 6: Systems and Matrices

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). Use this information to set up a matrix equation of the form $A X = B$ , which can be solved to determine the price each type of sale book. Solve this matrix equation to find the price of a book with a red sticker. Use the fact that for $A = \left[ \begin{array} { l l l } 1 & 2 & 2 \\ 1 & 3 & 2 \\ 1 & 2 & 3 \end{array} \right] , A ^ { - 1 } = \left[ \begin{array} { r r r } 5 & - 2 & - 2 \\ - 1 & 1 & 0 \\ - 1 & 0 & 1 \end{array} \right]$ .

Twice the water flow in the hot-water pipe is the same as three times the flow in the cold-water pipe. The combined flow is 1500 L/hr. What is the flow in each pipe?

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. y=(3x-2) +3=7

Solve the system. -2x-y+4z=-3 -2x+5y-9z=14 8x-8y+z=-31

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable. - 7x-9y-z =-24 x-3y-5z =-54 -2x+y+z =7

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. - x+by= x+ay=

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. x+ay= x+by=

Find the dimensions of a rectangular enclosure with perimeter 40 yd and area $91 \mathrm { yd } ^ { 2 }$

The sum of the squares of the digits of a positive two-digit number is 20, and the tens digit is 2 more than the units digit. Find the number.

Momma's ice cream shop sells three types of ice cream: soft-serve, chunky, and nonfat. Location I sells 11 gal of soft-serve, 100 gal of chunky, and 30 gal of nonfat ice cream each day. Location II sells 25 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, 21 gal of nonfat are sold each day, and 40 gal of nonfat are sold each day at Location III. Write a 3 × 3 matrix that shows the sales figures for the three locations, with the rows representing The three locations.

Provide an appropriate response. -When using the substitution or elimination method to solve a system of two equations, you end up with an equation stating $0 = 7$ What does this indicate to you about the system of equations?

Solve the system by substitution. -2x = 7y + 17 -31 = 3y - 3x

Graph the inequality. - $3 x+5 y \leq 15$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. x-y+z=5 x+y+z=-5 x+y-z=-3

Find the dimension of the matrix. - $\left[ \begin{array} { l l l } - 9 & 2 & 8\end{array} \right]$

Solve the linear programming problem. -A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average monthly salary of a counselor is $2400, and the average monthly salary of an aide is $1100. The camp can Accommodate up to 45 staff members and needs at least 30 to run properly. The camp must have at least 10 Aides and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the Camp hire to minimize cost?

Find the matrix product when possible. - $\left[ \begin{array} { l l l } - 5 & - 5 & 3 \end{array} \right] \left[ \begin{array} { r r r } 8 & 8 & 9 \\ 9 & - 7 & 7 \\ 6 & - 2 & - 2 \end{array} \right]$

Mike's Bait Shop sells three types of lures: discount, normal, and professional. Location I sells 29 discount lures, 100 regular lures, and 30 professional lures each day. Location II sells 10 discount lures and Location III sells 60 discount lures each day. Daily sales of regular lures are 90 at Location II and 120 at Location III. At Location II, 11 expert lures are sold each day, and 40 expert lures 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 lure for discount, normal, and professional lures are $\$ 3 , \$ 9$ , and $\$ 19$ , 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.

Provide an appropriate response. -Suppose that $\mathrm { A }$ and $\mathrm { B }$ are both matrices of dimension $\mathrm { r } \times \mathrm { s }$ . Under what conditions can both the product $\mathrm { AB }$ and the product BA be found?

Solve the system by elimination. 7x - 6y = 7 7x + 6y = 7