Solve for x by finding the determinant.

$$\left|\begin{array}{cc}x-2 & 2 \\ 2 & x+1\end{array}\right|=0$$

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

$$\dfrac{2 x-1}{x^5+2 x^4+4 x^3+8 x^2}$$

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

Evaluate the following determinant. See the earlier example.

$$\left|\begin{array}{cc}2 y & 3 x \\ y-1 & x^2\end{array}\right|$$

Determine values of the variables that will make the following equation true, if possible.

$$\left[\begin{array}{cc}w & 5 x \\ 2 y & z\end{array}\right]-3\left[\begin{array}{cc}w & x \\ 2 & -z\end{array}\right]=\left[\begin{array}{cc}4 & 2 \\ y-3 & -16\end{array}\right]$$

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

$$\left\{\begin{aligned} x^2+y^2 & =1 \\ y & =x^2-1\end{aligned}\right.$$

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.)

$$\left\{\begin{aligned} 2(z+3)-x+y & =z \\ -3(x-2 y)-1 & =5 z \\ \frac{x}{3}-(y-2 z) & =x\end{aligned}\right.$$

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.

$$2(A-3 B)$$

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

$$\left\{\begin{array}{r}2 x_1-3 x_3=7 \\ x_2-10 x_3=0 \\ 2 x_1-x_2+x_3=1\end{array}\right.$$

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.

$$\left\{\begin{array}{r}3 x+y=4 \\ -2 x+3 y=1\end{array}\right.$$

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

Lea ordered fruit baskets for three of her coworkers. One containing 5 apples, 2 oranges, and 1 mango cost $\$ 6.81$. Another containing 2 mangos, 8 oranges, and 3 apples cost $\$ 11.88$. The third contained 4 apples, 4 oranges, and 4 mangos and cost $\$ 11.04$. How much did each type of fruit cost?

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

Jael has $\$ 10,000$ she would like to invest. She has narrowed her options down to a certificate of deposit paying $5 \%$ annually, bonds paying $4 \%$ annually, and stocks with an expected annual rate of return of $13.5 \%$. If she wants to invest twice as much in the stocks as in the certificate of deposit and she wants to earn $\$ 1000$ in interest by the end of the year, how much should she invest in each type of investment?

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

A computer has 24 screws in its case. If there are 7 times more slotted screws than thumb screws, how many thumb screws are in the computer?

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

A movie brought in $\$ 740$ in ticket sales one day. Tickets during the day cost $\$ 5$ and tickets at night cost $\$ 7$. If 120 tickets were sold, how many were sold during the day?

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

Twelve years ago, Jim was twice as old as Kristin. Sixteen years ago, Jim was three times older than Kristin. How old is Jim?