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.

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

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.

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

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.)

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

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

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

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

$$\left\{\begin{aligned}3 x-2 y=6 \\ \frac{x^2}{4}+\frac{y^2}{9}=1\end{aligned}\right.$$

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.

$$\left\{\begin{array}{l}2 x-y=-12 \\ 3 x+y=-13\end{array}\right.$$

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.

$$3 A-B$$

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?

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.

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

Evaluate the following determinant. See the earlier example.

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

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

$$\left\{\begin{array}{r}14 x-5 y=7 \\ x+9 y=2\end{array}\right.$$

Write the form of the partial fraction decomposition of the following rational function. In this case, assume the degree of the numerator is less than the degree of the denominator. See the earlier examples.

$$f(x)=\dfrac{p(x)}{x^2-x-6}$$