Chapter 7

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 0.3 x+0.3 y+0.5 z=0.6 \\ 0.4 x+0.4 y+0.4 z=1.8 \\ 0.4 x+0.2 y+0.1 z=1.6 \end{array} $$

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions. $$ \begin{array}{c} 3 x-5 y=7 \\ x-2 y=3 \end{array} $$

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

For the following exercises, use a calculator to solve the system of equations with matrix inverses. $$ \begin{array}{l} 12.3 x-2 y-2.5 z=2 \\ 36.9 x+7 y-7.5 z=-7 \\ 8 y-5 z=-10 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} -\frac{1}{2} x+\frac{1}{2} y+\frac{1}{7} z=-\frac{53}{14} \\ \frac{1}{2} x-\frac{1}{2} y+\frac{1}{4} z=3 \\ \frac{1}{4} x+\frac{1}{5} y+\frac{1}{3} z=\frac{23}{15} \end{array} $$

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

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{x^{3}+6 x^{2}+5 x+9}{\left(x^{2}+1\right)^{2}} $$

For the following exercises, graph the system of inequalities. Label all points of intersection. $$ \begin{array}{l} x^{2}+3 y^{2}>16 \\ 3 x^{2}-y^{2}<1 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{aligned} &0.8 x+0.8 y+0.8 z=2.4\\\ &0.3 x-0.5 y+0.2 z=0\\\ &0.1 x+0.2 y+0.3 z=0.6 \end{aligned} $$

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions. $$ \begin{array}{c} 3 x-2 y=5 \\ -9 x+6 y=-15 \end{array} $$

For the following exercises, use the determinant function on a graphing utility. $$ \left|\begin{array}{rrrr} 1 & 0 & 2 & 1 \\ 0 & -9 & 1 & 3 \\ 3 & 0 & -2 & -1 \\ 0 & 1 & 1 & -2 \end{array}\right| $$

For the following exercises, use a calculator to solve the system of equations with matrix inverses. $$ \begin{array}{l} 0.5 x-3 y+6 z=-0.8 \\ 0.7 x-2 y=-0.06 \\ 0.5 x+4 y+5 z=0 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{aligned} -\frac{1}{2} x-\frac{1}{3} y+\frac{1}{4} z &=-\frac{29}{6} \\ \frac{1}{5} x+\frac{1}{6} y-\frac{1}{7} z &=\frac{431}{210} \\ -\frac{1}{8} x+\frac{1}{9} y+\frac{1}{10} z &=-\frac{49}{45} \end{aligned} $$

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

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

For the following exercises, graph the inequality. $$ \begin{array}{l} y \geq e^{x} \\ y \leq \ln (x)+5 \end{array} $$

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

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned} 0.1 x+0.2 y &=0.3 \\ -0.3 x+0.5 y &=1 \end{aligned} $$

For the following exercises, use the determinant function on a graphing utility. $$ \left|\begin{array}{rrrr} \frac{1}{2} & 1 & 7 & 4 \\ 0 & \frac{1}{2} & 100 & 5 \\ 0 & 0 & 2 & 2,000 \\ 0 & 0 & 0 & 2 \end{array}\right| $$

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

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

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

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{x^{2}+5 x+5}{(x+2)^{2}} $$

For the following exercises, graph the inequality. $$ \begin{array}{l} y \leq-\log (x) \\ y \leq e^{x} \end{array} $$

For the following exercises, solve the system for \(x, y,\) and \(z\). $$ \begin{array}{c} 5 x-3 y-\frac{z+1}{2}=\frac{1}{2} \\ 6 x+\frac{y-9}{2}+2 z=-3 \\ \frac{x+8}{2}-4 y+z=4 \end{array} $$

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned} -0.01 x+0.12 y &=0.62 \\ 0.15 x+0.20 y &=0.52 \end{aligned} $$

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 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 Gaussian elimination to solve the system. $$ \begin{array}{c} \frac{x-1}{4}-\frac{y+1}{4}+3 z=-1 \\ \frac{x+5}{2}+\frac{y+7}{4}-z=4 \\ x+y-\frac{z-2}{2}=1 \end{array} $$

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

For the following exercises, 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, find the solutions to the nonlinear equations with two variables. $$ \begin{array}{l} \frac{4}{x^{2}}+\frac{1}{y^{2}}=24 \\ \frac{5}{x^{2}}-\frac{2}{y^{2}}+4=0 \end{array} $$

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

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

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution. Two numbers add up to \(56 .\) One number is 20 less than the other.

For the following exercises, find the inverse of the given matrix. $$ \left[\begin{array}{rrrr} 1 & -2 & 3 & 0 \\ 0 & 1 & 0 & 2 \\ 1 & 4 & -2 & 3 \\ -5 & 0 & 1 & 1 \end{array}\right] $$

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

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

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{x^{2}+25}{\left(x^{2}+3 x+25\right)^{2}} $$

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

For the following exercises, solve the system for \(x, y,\) and \(z\). $$ \begin{array}{l} \frac{x-3}{6}+\frac{y+2}{2}-\frac{z-3}{3}=2 \\ \frac{x+2}{4}+\frac{y-5}{2}+\frac{z+4}{2}=1 \\ \frac{x+6}{2}-\frac{y-3}{2}+z+1=9 \end{array} $$

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned} 0.15 x+0.27 y &=0.39 \\ -0.34 x+0.56 y &=1.8 \end{aligned} $$

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution. Two numbers add up to 104 . If you add two times the first number plus two times the second number, your total is 208

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

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

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. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(A B\)

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{2 x^{3}+11 x^{2}+7 x+70}{\left(2 x^{2}+x+14\right)^{2}} $$

For the following exercises, find the solutions to the nonlinear equations with two variables. $$ \begin{array}{l} x^{2}-x y+y^{2}-2=0 \\ x+3 y=4 \end{array} $$

For the following exercises, solve the system for \(x, y,\) and \(z\). $$ \begin{array}{l} \frac{x-1}{3}+\frac{y+3}{4}+\frac{z+2}{6}=1 \\ 4 x+3 y-2 z=11 \\ 0.02 x+0.015 y-0.01 z=0.065 \end{array} $$

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. $$ \begin{aligned} -0.71 x+0.92 y &=0.13 \\ 0.83 x+0.05 y &=2.1 \end{aligned} $$