Solve the nonlinear system of equations.

$$\begin{aligned} & x^2+y^2=25 \\ & x+y=7 \end{aligned}$$

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.

$$\begin{gathered} x+2 y=5 \\ 2 x+y=-2 \end{gathered}$$

Find the partial fraction decomposition for the rational expression.

$$\frac{6 x^5+7 x^4-x^2+2 x}{3 x^2+2 x-1}$$

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

$$\begin{array}{r} x^2+y^2=5 \\ -3 x+4 y=2 \end{array}$$

Solve the given system by substitution.

$$\begin{aligned} &3 x-7 y=15 \\ &3 x+7 y=15 \end{aligned}$$

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

$$\left[\begin{array}{ll}-6 & 4 \\ -3 & 2\end{array}\right]$$

Graph the inequality.

$$(x-4)^2+(y+3)^2 \leq 9$$

Find the value of the determinant.

$$\left|\begin{array}{rrr}8 & -2 & -4 \\ 7 & 0 & 3 \\ 5 & -1 & 2\end{array}\right|$$

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.

$$\begin{aligned} & \frac{3}{x}+\frac{4}{y}=4 \\ & \frac{1}{x}+\frac{2}{y}=\frac{2}{3} \end{aligned}$$

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

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

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.

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

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

$$\left[\begin{array}{rr}5 & 10 \\ -3 & -6\end{array}\right]$$

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.

$$\begin{aligned} & 2 x+3 y+4 z=3 \\ & -4 x+2 y-6 z=2 \\ & 4 x+3 z=0 \end{aligned}$$

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

A cup of uncooked rice contains $15 \mathrm{~g}$ of protein and $810$ calories. A cup of uncooked soybeans contains $22.5 \mathrm{~g}$ of protein and $270$ calories. How many cups of each should be used for a meal containing $9.5 \mathrm{~g}$ of protein and $324$ calories?