Chapter 6

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

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

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

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

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

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{array}{c} x-y+z=-6 \\ 4 x+y+z=7 \end{array}

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. $$A=\left[\begin{array}{rrr} 5 & -3 & 2 \\ -5 & 3 & -2 \\ 1 & 0 & 1 \end{array}\right]$$

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

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

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{array}{r} 3 x-2 y+z=15 \\ x+4 y-z=11 \end{array}

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

Write an inequality that satisfies the description. Below the parabola with vertex ( \(0,1\) ) and \(x\) -intercepts \((-2,0)\) and \((2,0)\) \((-1,0)\) and \((1,0)\)

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

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

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

Determine whether each partial fraction decomposition is correct by graphing the left side and the right side of the equation on the same coordinate axes and observing whether the graphs coincide. $$\frac{4 x^{2}-3 x-4}{x^{3}+x^{2}-2 x}=\frac{2}{x}+\frac{-1}{x-1}+\frac{3}{x+2}$$

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+3 y+4 z &=3 \\ 6 x+3 y+8 z &=6 \\ 6 y-4 z &=1 \end{aligned}

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

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

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

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

Determine whether each partial fraction decomposition is correct by graphing the left side and the right side of the equation on the same coordinate axes and observing whether the graphs coincide. $$\frac{1}{(x-1)(x+2)}=\frac{1}{x-1}-\frac{1}{x+2}$$

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} 10 x+2 y-3 z &=0 \\ 5 x+4 y+6 z &=-1 \\ 6 y+3 z &=2 \end{aligned}

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

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

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

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

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{array}{r} \frac{1}{x}+\frac{1}{y}-\frac{1}{z}=\frac{1}{4} \\ \frac{2}{x}-\frac{1}{y}+\frac{3}{z}=\frac{9}{4} \\ -\frac{1}{x}-\frac{2}{y}+\frac{4}{z}=1 \end{array}

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

Solve each system by elimination. $$\begin{aligned}&6 x+7 y=-2\\\&7 x-6 y=26\end{aligned}$$

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

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

Determine whether each partial fraction decomposition is correct by graphing the left side and the right side of the equation on the same coordinate axes and observing whether the graphs coincide. $$\frac{2 x+4}{x^{2}(x-2)}=\frac{-2}{x}+\frac{-2}{x^{2}}+\frac{2}{x-2}$$

\begin{aligned} &\frac{3}{x}+\frac{2}{y}-\frac{1}{z}=\frac{11}{6}\\\ &\frac{1}{x}-\frac{1}{y}+\frac{3}{z}=-\frac{11}{12}\\\ &\frac{2}{x}+\frac{1}{y}+\frac{1}{z}=\frac{7}{12} \end{aligned}

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

Solve each system by elimination. $$\begin{array}{c}4 x-y=9 \\\\-8 x+2 y=-18\end{array}$$

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{array}{r}x+y \geq 0 \\\2 x-y \geq 3\end{array}$$

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

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

\(\begin{aligned} &\frac{2}{x}-\frac{2}{y}+\frac{1}{z}=-1\\\ &\frac{4}{x}+\frac{1}{y}-\frac{2}{z}=-9\\\ &\frac{1}{x}+\frac{1}{y}-\frac{3}{z}=-9 \end{aligned}\)

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

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

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{aligned}&x+y \leq 4\\\&x-2 y \geq 6\end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} &4 x+y=5\\\ &2 x+y=3 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{array}{c} x+3 y=-12 \\ 2 x-y=11 \end{array}$$