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.

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

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 $c_{51}$.

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 $c_{51}$.

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 what you have learned about systems of equations to answer these questions. Find values of a and b so that the system

$$\left\{ \begin{array} { l } { 3 x + 2 y = 5 } \\ { a x + 4 y = b } \end{array} \right.$$

has infinitely many solutions. Explain how the graphs of the equations will be related.

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

$$\left\{\begin{aligned} x-3 y+5 z & =6 \\ x-2 y+z & =5 \\ -4 x+5 y-6 z & =-17\end{aligned}\right.$$

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

$$\left\{\begin{array}{r}3 x+8 z=3 \\ x+3 z=1\end{array}\right.$$

Some systems of equations cannot be solved by using Cramer's Rule. a. Find the value of

$$\left| \begin{array} { l l } { a } & { b } \\ { f } & { y } \end{array} \right|$$

When is the value 0? b. Choose values for a, b, f, and g to make the determinant of the coefficient matrix 0. What type of system is formed?

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

$$\left\{\begin{aligned} 23 x-5 y & =244 \\ -31 x+29 y & =-84\end{aligned}\right.$$

Solve one of the systems by using a table, another system by sketching a graph, and the remaining system by using a graphing calculator. Explain why you chose each method.

a.

$$\begin{aligned} & y=4.3 x+1.2 \\ & y=-1.7 x-2.4 \end{aligned}$$

b.

$$\begin{aligned} & y=x \\ & y=-2 x+9 \end{aligned}$$

c.

$$\begin{aligned} & y=-x-5 \\ & y=3 x+1 \end{aligned}$$

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.

$$\left\{\begin{aligned} x+y-z & =1 \\ 3 x-4 y-5 z & =-1 \\ 6 x-3 y+z & =20\end{aligned}\right.$$

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

$$\left\{\begin{array}{c}x+2 y=-1 \\ y+3 z=7 \\ 2 x+5 z=21\end{array}\right.$$

Using a graphing calculator, we can solve systems that are too difficult to solve algebraically. Solve the system by graphing its equations on a graphing calculator and finding points of intersection to the nearest tenth.

$$\begin{aligned} &x^2-y^2=1 \\ &y=2^{x-3} \end{aligned}$$