Chapter 7

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{ll}2 x & x \\\11 & x\end{array}\right]=6$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & 2 & 1 & -3 \\ 0 & 1 & -3 & \frac{1}{2} \\ 0 & 0 & 0 & 4 \end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{1}{x(2 x+1)\left(3 x^{2}+4\right)}$$

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{lll} 2 & 1 & 2 \\ 5 & 10 & 5 \\ 3 & 6 & 3 \end{array}\right]$$

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

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 8 x-3 y+6 z &=-2 \\ 4 x+9 y+4 z &=18 \\ 12 x-3 y+8 z &=-2 \end{aligned}$$

Solve each system by elimination. $$\begin{aligned}3 x+4 y &=-1 \\\x-6 y &=-4\end{aligned}$$

Graph each inequality. $$y \geq \frac{1}{x+1}$$

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rrr}-2 & 0 & 1 \\\\-1 & 3 & x \\\5 & -2 & 0\end{array}\right]=3$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & 0 & -4 & \frac{3}{4} \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & -3 \end{array}\right]$$

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

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 5 & -3 & 2 \\ -5 & 3 & -2 \\ 1 & 0 & 1 \end{array}\right]$$

Perform each operation if possible. $$3\left[\begin{array}{rrr}1 & 0 & 3 \\ 0 & 1 & 2 \\ 1 & 0 & -3\end{array}\right]-4\left[\begin{array}{rrr}-1 & 0 & 0 \\ 0 & -1 & 3 \\ 2 & 0 & 1\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} &2 x+6 y-z=6\\\ &\begin{array}{l} 4 x-3 y+5 z=-5 \\ 6 x+9 y-2 z=11 \end{array} \end{aligned}$$

Solve each system by elimination. $$\begin{array}{c}-4 x-2 y=-2 \\\6 x+y=7\end{array}$$

Graph each inequality. $$y>\frac{1}{(x-1)^{2}}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} &x+y=5\\\ &x-y=-1 \end{aligned}$$

Solve each determinant equation for \(x\). $$\text { det }\left[\begin{array}{rrr}4 & 3 & 0 \\\2 & 0 & 1 \\\\-3 & x & -1\end{array}\right]=5$$

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

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{cc} \sqrt{2} & 0.5 \\ -17 & \frac{1}{2} \end{array}\right]$$

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

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

Solve each system by elimination. $$\begin{array}{r}3 x-y=-4 \\\x+3 y=12\end{array}$$

Graph each inequality. $$y<\frac{1}{x^{2}+1}$$

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{ll} \frac{2}{3} & 0.7 \\ 22 & \sqrt{3} \end{array}\right]$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{c} x+2 y=5 \\ 2 x+y=-2 \end{array}$$

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rrr}5 & 3 x & -3 \\\0 & 2 & -1 \\\4 & -1 & x\end{array}\right]=-7$$

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

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

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

Solve each system by elimination. $$\begin{aligned}&2 x-3 y=-7\\\&5 x+4 y=17\end{aligned}$$

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 0.1 & 0 & 0.1 \\ 0.2 & 0.1 & 0.3 \\ -0.1 & 0.1 & 0.1 \end{array}\right]$$

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rrr}2 x & 1 & -1 \\\0 & 4 & x \\\3 & 0 & 2\end{array}\right]=x$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{r} x+y=-3 \\ 2 x-5 y=-6 \end{array}$$

Find the partial fraction decomposition for each rational expression. $$\frac{-x^{4}-8 x^{2}+3 x-10}{(x+2)\left(x^{2}+4\right)^{2}}$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 3 x+2 y-z &=-1 \\ 3 y+z &=12 \\ x-3 z &=-3 \end{aligned}$$

Solve each system by elimination. $$\begin{aligned}&4 x+3 y=-1\\\&2 x+5 y=3\end{aligned}$$

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

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{lll} \frac{1}{2} & \frac{1}{4} & \frac{1}{3} \\ 0 & \frac{1}{4} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{3} \end{array}\right]$$

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{lll}x & x & 2 \\\0 & 2 & 2 \\\0 & 0 & 3 x\end{array}\right]=96$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} &3 x-2 y=4\\\ &3 x+y=-2 \end{aligned}$$

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

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 2 x+y-z &=-4 \\ y+2 z &=12 \\ 2 x-z &=-4 \end{aligned}$$

Solve each system by elimination. $$\begin{array}{r}5 x+7 y=6 \\\10 x-3 y=46\end{array}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} &2 x-y=-8\\\ &3 x+y=-2 \end{aligned}$$

Evaluate each determinant. $$\operatorname{det}\left[\begin{array}{rrrr}3 & -6 & 5 & -1 \\\0 & 2 & -1 & 3 \\\\-6 & 4 & 2 & 0 \\\\-7 & 3 & 1 & 1\end{array}\right]$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} &2 x-3 y=10\\\ &2 x+2 y=5 \end{aligned}$$

Find the partial fraction decomposition for each rational expression. $$\frac{5 x^{5}+10 x^{4}-15 x^{3}+4 x^{2}+13 x-9}{x^{3}+2 x^{2}-3 x}$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} &x-y+z=-6\\\ &x+y+3 z=10 \end{aligned}$$

Solve each system by elimination. $$\begin{aligned}12 x-5 y &=9 \\\3 x-8 y &=-18\end{aligned}$$