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.

$$\left\{\begin{array}{l}4 x-2 y=3 \\ -2 x+y=-7\end{array}\right.$$

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

$$\left\{\begin{array}{r}x^2+y^2=9 \\ x^2+y^2-2 x-3=1\end{array}\right.$$

Evaluate the following matrix product, if possible.

$$\left[\begin{array}{ccc}7 & 1 & -1\end{array}\right]\left[\begin{array}{cc}1 & 6 \\ 2 & 1 \\ -3 & -3\end{array}\right]$$

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

$$\dfrac{2 x-1}{x^5+6 x^4+12 x^3+8 x^2}$$

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

Find the inverse of the following matrices, if possible. See the earlier examples.

$$\left[\begin{array}{cc}0 & 4 \\ -5 & -1\end{array}\right]$$

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

$$\left\{\begin{array}{l}(x-2)^2+y^2=4 \\ (x+2)^2+y^2=4\end{array}\right.$$

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.

$$\left\{\begin{aligned} 4 x-y & =-1 \\ -8 x+2 y & =2\end{aligned}\right.$$

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

$$\dfrac{2 x-1}{x^3(x-2)}$$

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

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

$$\left[\begin{array}{lll}4 x & 2 y^2 & z\end{array}\right]=\left[\begin{array}{l}12 \\ 18 \\ -2\end{array}\right]$$

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.)

$$\left\{\begin{aligned} \frac{12 x-1}{5}+\frac{y}{2} & =\frac{3 z}{2} \\ y-(x+3 z) & =-(1-y) \\ 2 x-2-z-2 y & =7 x\end{aligned}\right.$$

Solve for x by finding the determinant.

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

Find the minimum and maximum values of the given functions, subject to the given constraints. See the earlier examples.

Objective Function

$$f(x, y)=6 x+8 y$$

Constraints

$$x \geq 0 ; y \geq 0$$

$$4 x+y \leq 16$$

$$x+3 y \leq 15$$

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

$$\left[\begin{array}{lll}a & 2 b & c\end{array}\right]+3\left[\begin{array}{lll}a & 2 & -c\end{array}\right]=\left[\begin{array}{lll}8 & 2 & 2\end{array}\right]$$

Find the minimum and maximum values of the given functions, subject to the given constraints. See the earlier examples.

Objective Function

$$f(x, y)=3 x+7 y$$

Constraints

$$0 \leq x \leq 8$$

$$0 \leq y \leq 6$$

$$7 x+10 y \leq 50$$