Chapter 7

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors. $$ \frac{x^{3}-5 x^{2}+12 x+144}{x^{2}\left(x^{2}+12 x+36\right)} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{l} x^{2}+y^{2}=25 \\ x^{2}-y^{2}=36 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{aligned} \frac{1}{2} x-\frac{1}{5} y+\frac{2}{5} z &=-\frac{13}{10} \\ \frac{1}{4} x-\frac{2}{5} y-\frac{1}{5} z &=-\frac{7}{20} \\ -\frac{1}{2} x-\frac{3}{4} y-\frac{1}{2} z &=-\frac{5}{4} \end{aligned} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 4 x-3 y=-3 \\ 2 x+6 y=-4 \end{array} $$

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} -3 x-4 y=9 \\ 12 x+4 y=-6 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} \frac{3}{4} x-\frac{3}{5} y=4 \\ \frac{1}{4} x+\frac{2}{3} y=1 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B A\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{4 x^{2}+6 x+11}{(x+2)\left(x^{2}+x+3\right)} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}+y^{2}=1 \\ y^{2}=x^{2} \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} -\frac{1}{3} x-\frac{1}{2} y-\frac{1}{4} z=\frac{3}{4} \\ -\frac{1}{2} x-\frac{1}{4} y-\frac{1}{2} z=2 \\ -\frac{1}{4} x-\frac{3}{4} y-\frac{1}{2} z=-\frac{1}{2} \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 4 x-5 y=7 \\ -3 x+9 y=0 \end{array} $$

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} -2 x+3 y=\frac{3}{10} \\ -x+5 y=\frac{1}{2} \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} \frac{1}{4} x-\frac{2}{3} y=-1 \\ \frac{1}{2} x+\frac{1}{3} y=3 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(C A\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{4 x^{2}+9 x+23}{(x-1)\left(x^{2}+6 x+11\right)} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{aligned} 16 x^{2}-9 y^{2}+144 &=0 \\ y^{2}+x^{2} &=16 \end{aligned} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} \frac{1}{2} x-\frac{1}{4} y+\frac{3}{4} z=0 \\ \frac{1}{4} x-\frac{1}{10} y+\frac{2}{5} z=-2 \\ \frac{1}{8} x+\frac{1}{5} y-\frac{1}{8} z=2 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 4 x+10 y=180 \\ -3 x-5 y=-105 \end{array} $$

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} \frac{8}{5} x-\frac{4}{5} y=\frac{2}{5} \\ -\frac{8}{5} x+\frac{1}{5} y=\frac{7}{10} \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{lll|l} 1 & 0 & 0 & 31 \\ 0 & 1 & 1 & 45 \\ 0 & 0 & 1 & 87 \end{array}\right] $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B C\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{-2 x^{2}+10 x+4}{(x-1)\left(x^{2}+3 x+8\right)} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{aligned} 3 x^{2}-y^{2} &=12 \\ (x-1)^{2}+y^{2} &=1 \end{aligned} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{aligned} \frac{4}{5} x-\frac{7}{8} y+\frac{1}{2} z &=1 \\ -\frac{4}{5} x-\frac{3}{4} y+\frac{1}{3} z &=-8 \\ -\frac{2}{5} x-\frac{7}{8} y+\frac{1}{2} z &=-5 \end{aligned} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 8 x-2 y=-3 \\ -4 x+6 y=4 \end{array} $$

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} \frac{1}{2} x+\frac{1}{5} y=-\frac{1}{4} \\ \frac{1}{2} x-\frac{3}{5} y=-\frac{9}{4} \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{lll|r} 1 & 0 & 1 & 50 \\ 1 & 1 & 0 & 20 \\ 0 & 1 & 1 & -90 \end{array}\right] $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(A^{2}\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{x^{2}+3 x+1}{(x+1)\left(x^{2}+5 x-2\right)} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{aligned} 3 x^{2}-y^{2} &=12 \\ (x-1)^{2}+y^{2} &=4 \end{aligned} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} -\frac{1}{3} x-\frac{1}{8} y+\frac{1}{6} z=-\frac{4}{3} \\ -\frac{2}{3} x-\frac{7}{8} y+\frac{1}{3} z=-\frac{23}{3} \\ -\frac{1}{3} x-\frac{5}{8} y+\frac{5}{6} z=0 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} x+2 y-4 z=-1 \\ 7 x+3 y+5 z=26 \\ -2 x-6 y+7 z=-6 \end{array} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix.$$ \begin{array}{l} 3 x-2 y+5 z=21 \\ 5 x+4 y=37 \\ x-2 y-5 z=5 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{lll|l} 1 & 2 & 3 & 4 \\ 0 & 5 & 6 & 7 \\ 0 & 0 & 8 & 9 \end{array}\right] $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B^{2}\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{4 x^{2}+17 x-1}{(x+3)\left(x^{2}+6 x+1\right)} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{aligned} 3 x^{2}-y^{2} &=12 \\ x^{2}+y^{2} &=16 \end{aligned} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} -\frac{1}{4} x-\frac{5}{4} y+\frac{5}{2} z=-5 \\ -\frac{1}{2} x-\frac{5}{3} y+\frac{5}{4} z=\frac{55}{12} \\ -\frac{1}{3} x-\frac{1}{3} y+\frac{1}{3} z=\frac{5}{3} \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} -5 x+2 y-4 z=-47 \\ 4 x-3 y-z=-94 \\ 3 x-3 y+2 z=94 \end{array} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$ \begin{array}{l} 3 x-2 y+5 z=21 \\ 5 x+4 y=37 \\ x-2 y-5 z=5 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{rrr|r} -0.1 & 0.3 & -0.1 & 0.2 \\ -0.4 & 0.2 & 0.1 & 0.8 \\ 0.6 & 0.1 & 0.7 & -0.8 \end{array}\right] $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(C^{2}\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{4 x^{2}}{(x+5)\left(x^{2}+7 x-5\right)} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}-y^{2}-6 x-4 y-11=0 \\ -x^{2}+y^{2}=5 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{aligned} \frac{1}{40} x+\frac{1}{60} y+\frac{1}{80} z &=\frac{1}{100} \\ -\frac{1}{2} x-\frac{1}{3} y-\frac{1}{4} z &=-\frac{1}{5} \\ \frac{3}{8} x+\frac{3}{12} y+\frac{3}{16} z &=\frac{3}{20} \end{aligned} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 4 x+5 y-z=-7 \\ -2 x-9 y+2 z=8 \\ 5 y+7 z=21 \end{array} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$ \begin{array}{l} 6 x-5 y-z=31 \\ -x+2 y+z=-6 \\ 3 x+3 y+2 z=13 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{r} -2 x+3 y-2 z=3 \\ 4 x+2 y-z=9 \\ 4 x-8 y+2 z=-6 \end{array} $$

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B^{2} A^{2}\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{4 x^{2}+5 x+3}{x^{3}-1} $$