Use a system of equations to solve the following problems. See the earlier examples.

A tour organizer is planning on taking a group of 40 people to a musical. Balcony tickets cost $\$ 29.95$ and regular tickets cost $\$ 19.95$, and she collects a total of $\$ 1048.00$ from her group to buy the tickets. How many people chose to sit in the balcony?

Use a system of equations to solve the following problems. See the earlier examples.

Karen empties out her purse and finds 45 loose coins, consisting entirely of nickels and pennies. If the total value of the coins is $\$ 1.37$, how many nickels and how many pennies does she have?

Use graphing to guess the real solution(s) of the following systems, and then verify your answer algebraically. See the earlier examples.

$$\left\{\begin{array}{l}(x-1)^2+(y-6)^2=9 \\ (x-1)^2+(y+1)^2=16\end{array}\right.$$

Evaluate the following determinant.

$$\left|\begin{array}{cccc}-2 & -1 & -3 & 0 \\ 3 & 3 & 1 & 5 \\ 4 & 0 & 0 & 1 \\ 2 & 0 & 1 & 0\end{array}\right|$$

Construct the augmented matrix that corresponds to the following systems of equation. See the earlier example. (Answers may appear in slightly different, but equivalent, forms.)

$$\left\{\begin{aligned} 5 x+\frac{y-z}{2} & =3 \\ 7(z-x)+y-2 & =0 \\ x-(4-z) & =y\end{aligned}\right.$$

Find the minimum and maximum values of the given functions, subject to the given constraints. See the earlier examples.

Objective Function

$$f(x, y)=9 x+7 y$$

Constraints

$$0 \leq x \leq 20$$

$$0 \leq y \leq 10$$

$$6 x+12 y \leq 140$$

Evaluate the following determinant. See the earlier example.

$$\left|\begin{array}{ll}w & x \\ y & z\end{array}\right|$$

Given

$$A=\left[\begin{array}{cc}3 & -2 \\ 1 & 0 \\ 0 & 5\end{array}\right], B=\left[\begin{array}{cc}4 & -5 \\ 3 & 0 \\ -2 & 2\end{array}\right], C=\left[\begin{array}{cc}2 & -1 \\ 6 & 10 \\ -3 & 7\end{array}\right]$$

, and

$$D=\left[\begin{array}{ccc}3 & 2 & 5 \\ -2 & -4 & 1\end{array}\right]$$

, determine the following, if possible. See the earlier examples.

$$3 C-A$$

Use the method of substitution to solve the following system of equation. If a system is dependent, express the solution set in terms of one of the variables. See the earlier examples.

$$\left\{\begin{array}{r}5 x+2 y=8 \\ 2 x+y=6\end{array}\right.$$

Match each rational expression with the form of its decomposition. The decompositions are labeled a-h.

$$\dfrac{2 x-1}{x^4-16}$$

a. $\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{x^3}+\dfrac{D}{x-2}+\dfrac{E}{(x-2)^2}$

b. $\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{x+2}+\dfrac{D}{x-2}$

c. $\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{x^3}+\dfrac{D}{x-2}$

d. $\dfrac{A}{(x+2)}+\dfrac{B}{(x+2)^2}+\dfrac{C}{(x+2)^3}+\dfrac{D}{x-2}$

e. $\dfrac{A x+B}{x^2+4}+\dfrac{C}{x+2}+\dfrac{D}{x-2}$

f. $\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{C}{x+2}+\dfrac{D x+E}{x^2+4}$

g. $\dfrac{A}{x+2}+\dfrac{B}{(x+2)^2}+\dfrac{C}{(x+2)^3}+\dfrac{D}{x}+\dfrac{E}{x^2}$

h. $\dfrac{A}{x}+\dfrac{B}{x-2}+\dfrac{C}{(x-2)^2}$

Write each of the following systems of equation as a single matrix equation. See the earlier example.

$$\left\{\begin{array}{l}x_3=x_2 \\ x_2=x_1 \\ x_1=x_3\end{array}\right.$$

Construct the constraints and graph the feasible regions for the following situations. See the earlier example.

A plane carrying relief food and water can carry a maximum of 50,000 pounds, and is limited in space to carrying no more than 6000 cubic feet. Each container of water weighs 60 pounds and takes up 1 cubic foot, and each container of food weighs 50 pounds and takes up 10 cubic feet. What is the region of constraint for the number of containers of food and water that the plane can carry?

Let

$$A=\left[\begin{array}{cc}4 & -1 \\ 0 & 3 \\ 9 & -5\end{array}\right]$$

. Determine the following, if possible:

a. The order of $A$.

b. The value of $a_{12}$.

c. The value of $a_{23}$.Let

$$C=\left[\begin{array}{cc}1 & 0 \\ 5 & -3 \\ 2 & 9 \\ \pi & e \\ 10 & -7\end{array}\right]$$

. Determine the following, if possible:

a. The order of $C$.

b. The value of $c_{23}$.

c. The value of $c_{51}$.

Use any convenient method to solve the following systems of equation. If a system is dependent, express the solution set in terms of one or more of the variables, as appropriate.

$$\left\{\begin{aligned} 3 x-y+z & =2 \\ -x+y-2 z & =-4 \\ -6 x+2 y-2 z & =-7\end{aligned}\right.$$