The current and estimated resident populations (in percent) of blacks and Hispanics in the United States for the years $1990-2050$ are modeled by the linear functions defined below.

$$\begin{array}{ll} y=.0515 x+12.3 & \text { Blacks } \\ y=.255 x+9.01 & \text { Hispanics } \end{array}$$

In the case, $x$ represents the number of years since 1990.

($c$) Use a calculator graph of the system to support your algebraic solution.

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

$$[-9]$$

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.

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

Solve the given system by substitution.

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

Find the partial fraction decomposition for the rational expression.

$$\frac{x^3+2}{x^3-3 x^2+2 x}$$

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

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

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

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.

$$\begin{array}{r} x+5 y+3 z=9 \\ 2 x+9 y+7 z=5 \end{array}$$

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.

Find the value of the determinant.

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

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

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

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.

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

Solve the nonlinear system of equations.

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

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

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