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

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

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.

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

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 $c_{51}$.

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.

$$3 A-B$$

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.

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

Evaluate the following determinant. See the earlier example.

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

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

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

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

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

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 $c_{51}$.