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

Eliza's mother is 20 years older than Eliza, but 3 years younger than Eliza's father. Eliza's father is 7 years younger than three times Eliza's age. How old is Eliza?

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?

Evaluate the following determinant. See the earlier example.

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

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

$$0 \leq x \leq 20$$

$$0 \leq y \leq 10$$

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

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.

$$3 C-A$$

Evaluate the following determinant.

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

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 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.

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

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.)

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

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.

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

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.

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

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.

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

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

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