91Ó°ÊÓ

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

For the following exercises, find the inverse of the given matrix. $$\left[\begin{array}{rrrr}{-1} & {0} & {2} & {5} \\ {0} & {0} & {0} & {2} \\\ {0} & {2} & {-1} & {0} \\ {1} & {-3} & {0} & {1}\end{array}\right]$$

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: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{ll}{1} & {0} \\ {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{-2} & {3} & {4} \\ {-1} & {1} & {-5}\end{array}\right], C=\left[\begin{array}{rr}{0.5} & {0.1} \\ {1} & {0.2} \\\ {-0.5} & {0.3}\end{array}\right], D=\left[\begin{array}{rrr}{1} & {0} & {-1} \\ {-6} & {7} & {5} \\ {4} & {2} & {1}\end{array}\right] $$ $$ (A B) C $$

For the following exercises, solve the system for \(x, y,\) and \(z\) $$ \begin{array}{l}{\frac{x+4}{7}-\frac{y-1}{6}+\frac{z+2}{3}=1} \\\ {\frac{x-2}{4}+\frac{y+1}{8}-\frac{z+8}{12}=0} \\\ {\frac{x+6}{3}-\frac{y+2}{3}+\frac{z+4}{2}=3}\end{array} $$

Find the solutions to the nonlinear equations with two variables. $$ \begin{array}{r} \frac{4}{x^{2}}+\frac{1}{y^{2}}=24 \\ \frac{5}{x^{2}}-\frac{2}{y^{2}}+4=0 \end{array} $$

For the following exercises, use the determinant function on a graphing utility. \(\left|\begin{array}{llll}1 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 \\ 4 & 5 & 6 & 0 \\\ 7 & 8 & 9 & 0\end{array}\right|\)

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$\frac{x^{3}+2 x^{2}+4 x}{\left(x^{2}+2 x+9\right)^{2}}$$

Find the decomposition of the partial fraction for the irreducible repeating quadratic factor. \(\frac{x^{3}+2 x^{2}+4 x}{\left(x^{2}+2 x+9\right)^{2}}\)

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned} 0.5 x+0.3 y &=4 \\ 0.25 x-0.9 y &=0.46 \end{aligned} $$

For the following exercises, find the solutions to the nonlinear equations with two variables. $$\frac{6}{x^{2}}-\frac{1}{y^{2}}=8$$ $$\frac{1}{x^{2}}-\frac{6}{y^{2}}=\frac{1}{8}$$