Chapter 6

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

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

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

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

Perform each operation if possible. $$\left[\begin{array}{rr} 12 & -5 \\ 10 & 3 \end{array}\right]-\left[\begin{array}{rr} 6 & 9 \\ -2 & 0 \end{array}\right]$$

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

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

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

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rr|r} 1 & -5 & 6 \\ 0 & 0 & 0 \end{array}\right]$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}0.4 & -0.8 & 0.6 \\\0.3 & 0.9 & 0.7 \\\3.1 & 4.1 & -2.8\end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rrr} -8 & 4 & 0 \\ 2 & 5 & 0 \end{array}\right]+\left[\begin{array}{ll} 6 & 3 \\ 8 & 9 \end{array}\right]$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \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 substitution. $$\begin{aligned}&3 y=5 x+6\\\&x+y=2\end{aligned}$$

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

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

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{aligned} &2 x+6 y-z=6\\\ &4 x-3 y+5 z=-5\\\ &6 x+9 y-2 z=11 \end{aligned}

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

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}-0.3 & -0.1 & 0.9 \\\2.5 & 4.9 & -3.2 \\\\-0.1 & 0.4 & 0.8\end{array}\right]$$

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

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

Use the concepts of this section to work. Which one of the following is a description of the graph of the inequality \((x-5)^{2}+(y-2)^{2}<4 ?\) A. The region inside a circle with center \((-5,-2)\) and radius 2 B. The region inside a circle with center \((5,2)\) and radius 2 C. The region inside a circle with center \((-5,-2)\) and radius 4 D. The region outside a circle with center \((5,2)\) and radius 4

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} -1 & -1 & -1 \\ 4 & 5 & 0 \\ 0 & 1 & -3 \end{array}\right]$$

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

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{aligned} &x-z=2\\\ &x+y=-3\\\ &y-z=1 \end{aligned}

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

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}17 & -4 & 3 \\\11 & 5 & -15 \\\7 & -9 & 23\end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rrrr} 9 & 4 & 1 & -2 \\ 5 & -6 & 3 & 4 \\ 2 & -5 & 1 & 2 \end{array}\right]-\left[\begin{array}{rrrr} -2 & 5 & 1 & 3 \\ 0 & 1 & 0 & 2 \\ -8 & 3 & 2 & 1 \end{array}\right]+\left[\begin{array}{rrrr} 2 & 4 & 0 & 3 \\ 4 & -5 & 1 & 6 \\ 2 & -3 & 0 & 8 \end{array}\right]$$

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

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 2 & 0 & 4 \\ 3 & 1 & 5 \\ -1 & 1 & -2 \end{array}\right]$$

Use the concepts of this section to work. Which one of the given inequalities satisfies the following description: the region outside a circle centered at the origin, with \(x\) -intercepts \((4,0)\) and \((-4,0) ?\) A. \(x^{2}+y^{2}>4\) B. \((x-4)^{2}+y^{2}>16\) C. \(x^{2}+y^{2}<16\) D. \(x^{2}+y^{2}>16\)

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

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{aligned} &x+z=4\\\ &x+y=4\\\ &y+z=4 \end{aligned}

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

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rr}5 & x \\\\-3 & 2\end{array}\right]=6$$

Perform each operation if possible. $$\left[\begin{array}{rrr} 6 & -2 & 4 \\ -2 & 5 & 8 \\ 1 & 0 & 2 \end{array}\right]+\left[\begin{array}{rrr} 3 & 0 & 8 \\ 1 & -2 & 4 \\ 6 & 9 & -2 \end{array}\right]-\left[\begin{array}{rrr} -4 & 2 & 1 \\ 0 & 3 & -2 \\ 4 & 2 & 0 \end{array}\right]$$

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

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{ccc} -0.4 & 0.1 & 0.2 \\ 0 & 0.6 & 0.8 \\ 0.3 & 0 & -0.2 \end{array}\right]$$

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

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & 2 & -1 & 5 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Perform each operation if possible. $$2\left[\begin{array}{rr} 2 & -1 \\ 5 & 1 \\ 0 & 3 \end{array}\right]+\left[\begin{array}{rr} 5 & 0 \\ 7 & -3 \\ 1 & 1 \end{array}\right]-\left[\begin{array}{rr} 9 & -4 \\ 4 & 4 \\ 1 & 6 \end{array}\right]$$

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rl}-0.5 & 2 \\\x & x\end{array}\right]=0$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{aligned} 3 x+2 y-z &=-1 \\ 3 y+z &=12 \\ x-3 z &=-3 \end{aligned}

Solve each system by substitution. $$\begin{aligned}-3 x+2 y &=-10 \\\9 x-6 y &=30\end{aligned}$$

Write an inequality that satisfies the description. Outside the circle with radius 3 and center ( \(0,0\) )

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 0.8 & 0.2 & 0.1 \\ -0.2 & 0 & 0.3 \\ 0 & 0 & 0.5 \end{array}\right]$$

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

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \begin{aligned} 2 x+y-z &=-4 \\ y+2 z &=12 \\ 2 x-z &=-4 \end{aligned}

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & -1 & 2 & 8 \\ 0 & 1 & -4 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

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

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