Use the Gauss-Jordan method to solve the system of equations. For systems in three variables with infinitely many solutions, give the solution with $z$ arbitrary; for any such equations with four variables, let $w$ be the arbitrary variable.

$$\begin{aligned} &2 x-3 y=10 \\ &2 x+2 y=5 \end{aligned}$$

Graph the inequality.

When graphing $y \leq 3 x-6$, would you shade above or below the line $y=3 x-6$ ? Explain your answer.

Solve the given system by substitution.

$$\begin{aligned} &4 y=2 x-4 \\ &x-y=4 \end{aligned}$$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.

$$\begin{array}{r} x^2+y^2=10 \\ 2 x^2-y^2=17 \end{array}$$

Find two numbers such that their sum is $-1$ and the sum of their squares is $61$ .

Find the value of the determinant.

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

Find the inverse, if it exists, for the matrix.

$$\left[\begin{array}{rrr}1 & 0 & 1 \\ 0 & -1 & 0 \\ 2 & 1 & 1\end{array}\right]$$

Find the size of the matrix. Identify any square, col

$$\left[\begin{array}{l}2 \\ 4\end{array}\right]$$

Solve the given system by substitution.

$$\begin{aligned} &3 y=5 x+6 \\ &x+y=2 \end{aligned}$$

Find the partial fraction decomposition for the rational expression.

$$\frac{x^3+4}{9 x^3-4 x}$$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.

$$\begin{aligned} & x^2+y^2=8 \\ & x^2-y^2=0 \end{aligned}$$

This exercise has introduced methods for solving systems of equations, including substitution and elimination, and matrix methods such as the Gauss-Jordan method and Cramer's rule. Use each method at least once when solving the systems. Include solutions with nonreal complex number components. For systems with infinitely many solutions, write the solution set using an arbitrary variable.

$$\begin{aligned} & -5 x+2 y+z=5 \\ & -3 x-2 y-z=3 \\ & -x+6 y=1 \end{aligned}$$

Write a system of linear equations, and then use the system to solve the problem.

The Waputi Indians make woven blankets, rugs, and skirts. Each blanket requires $24 \mathrm{hr}$ for spinning the yarn, $4 \mathrm{hr}$ for dyeing the yarn, and $15 \mathrm{hr}$ for weaving. Rugs require $30$,$5$ , and $18 \mathrm{hr}$ and skirts $12$,$3$ , and $9 \mathrm{hr}$, respectively. If there are $306$,$59$ , and $201 \mathrm{hr}$ available for spinning, dyeing, and weaving, respectively, how many of each item can be made?

(Hint: Simplify the equations you write, if possible, before solving the system.)

Graph the inequality.

In your own words, explain how to determine whether the boundary of the graph of an inequality is solid or dashed.