Chapter 7

Solve each system by substitution. $$\begin{aligned}&4 x+5 y=7\\\&9 y=31+2 x\end{aligned}$$

Graph each inequality. $$x<3+2 y$$

Find the partial fraction decomposition for each rational expression. $$\frac{x+1}{x^{2}(1-x)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 0.8 & -0.3 \\ 0.5 & -0.2 \end{array}\right]$$

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{rr|r} 1 & -5 & -18 \\ 6 & 2 & 20 \end{array}\right]$$

Explain to a friend in your own words how to multiply a matrix by a scalar.

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 4 x-3 y+z &=9 \\ 3 x+2 y-2 z &=4 \\ x-y+3 z &=5 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}&2 x+6 y=-18\\\&5 y=-29+3 x\end{aligned}$$

Find each determinant. $$\text { det }\left[\begin{array}{ccc}10 & 2 & 1 \\\\-1 & 4 & 3 \\\\-3 & 8 & 10\end{array}\right]$$

Graph each inequality. $$5 x \leq 4 y-2$$

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{lll|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -2 \end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{4 x^{2}-x-15}{x(x+1)(x-1)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rrr}6 & -9 & 2 \\ 4 & 1 & 3\end{array}\right]+\left[\begin{array}{rrr}-8 & 2 & 5 \\ 6 & -3 & 4\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x+2 y+3 z &=8 \\ 3 x-y+2 z &=5 \\ -2 x-4 y-6 z &=5 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}&3 x-7 y=15\\\&3 x+7 y=15\end{aligned}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}7 & -1 & 1 \\\1 & -7 & 2 \\\\-2 & 1 & 1\end{array}\right]$$

Graph each inequality. $$2 x>3-4 y$$

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{lll|l} 1 & 0 & 1 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{2 x+1}{(x+2)^{3}}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rr}9 & 4 \\ -8 & 2\end{array}\right]+\left[\begin{array}{cc}-3 & 2 \\ -4 & 7\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 3 x-2 y-8 z &=1 \\ 9 x-6 y-24 z &=-2 \\ x-y+z &=1 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}&3 y=5 x+6\\\&x+y=2\end{aligned}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}1 & -2 & 3 \\\0 & 0 & 0 \\\1 & 10 & -12\end{array}\right]$$

Graph each inequality. $$y

Find the partial fraction decomposition for each rational expression. $$\frac{x^{2}}{x^{2}+2 x+1}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rrr} 1 & 0 & 1 \\ 2 & 1 & 3 \\ -1 & 1 & 1 \end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rr}-6 & 8 \\ 0 & 0\end{array}\right]-\left[\begin{array}{rr}0 & 0 \\ -4 & -2\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x+4 y-z &=6 \\ 2 x-y+z &=3 \\ 3 x+2 y+3 z &=16 \end{aligned}$$

Solve each system by substitution. $$\begin{array}{c}2 x-7 y=8 \\\\-3 x+\frac{21}{2} y=5\end{array}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}2 & 3 & 0 \\\1 & 9 & 0 \\\\-1 & -2 & 0\end{array}\right]$$

Graph each inequality. $$y \leq x^{2}-4$$

Find the partial fraction decomposition for each rational expression. $$\frac{3}{x^{2}+4 x+3}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rrr} -2 & 1 & 0 \\ 1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rr}1 & -4 \\ 2 & -3 \\ -8 & 4\end{array}\right]-\left[\begin{array}{rr}-6 & 9 \\ -2 & 5 \\ -7 & -12\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 4 x-y+3 z &=-2 \\ 3 x+5 y-z &=15 \\ -2 x+y+4 z &=14 \end{aligned}$$

Solve each system by substitution. $$\begin{array}{r}0.6 x-0.2 y=2 \\\\-1.2 x+0.4 y=3\end{array}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}3 & 3 & -1 \\\2 & 6 & 0 \\\\-6 & -6 & 2\end{array}\right]$$

Graph each inequality. $$y \leq 1-x^{2}$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rr|r} 1 & 2 & 3 \\ 0 & 1 & -1 \end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{2 x^{5}+3 x^{4}-3 x^{3}-2 x^{2}+x}{2 x^{2}+5 x+2}$$

Under what condition will the inverse of a square matrix not exist?

Perform each operation if possible. $$\left[\begin{array}{rr}6 & -2 \\ 5 & 4\end{array}\right]+\left[\begin{array}{rr}-1 & 7 \\ 7 & -4\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 5 x+y-3 z &=-6 \\ 2 x+3 y+z &=5 \\ -3 x-2 y+4 z &=3 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}x-2 y &=4 \\\\-2 x+4 y &=-8\end{aligned}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}5 & -3 & 2 \\\\-5 & 3 & -2 \\\1 & 0 & 1\end{array}\right]$$

Graph each inequality. $$y<2-3 x^{2}$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rr|r} 1 & -1 & 2 \\ 0 & 1 & 0 \end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{6 x^{5}+7 x^{4}-x^{2}+2 x}{3 x^{2}+2 x-1}$$