Chapter 7

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}+y^{2}-6 y=7 \\ x^{2}+y=1 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{aligned} &0.1 x-0.2 y+0.3 z=2\\\ &0.5 x-0.1 y+0.4 z=8\\\ &0.7 x-0.2 y+0.3 z=8 \end{aligned} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 4 x-3 y+4 z=10 \\ 5 x-2 z=-2 \\ 3 x+2 y-5 z=-9 \end{array} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$ \begin{array}{l} 6 x-5 y+2 z=-4 \\ 2 x+5 y-z=12 \\ 2 x+5 y+z=12 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{aligned} x+y-4 z &=-4 \\ 5 x-3 y-2 z &=0 \\ 2 x+6 y+7 z &=30 \end{aligned} $$

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: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(A^{2} B^{2}\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{-5 x^{2}+18 x-4}{x^{3}+8} $$

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{c} x^{2}+y^{2}=6 \\ x y=1 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 0.2 x+0.1 y-0.3 z=0.2 \\ 0.8 x+0.4 y-1.2 z=0.1 \\ 1.6 x+0.8 y-2.4 z=0.2 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 4 x-2 y+3 z=6 \\ -6 x+y=-2 \\ 2 x+7 y+8 z=24 \end{array} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$ \begin{array}{l} 4 x-2 y+3 z=-12 \\ 2 x+2 y-9 z=33 \\ 6 y-4 z=1 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{c} 2 x+3 y+2 z=1 \\ -4 x-6 y-4 z=-2 \\ 10 x+15 y+10 z=5 \end{array} $$

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: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \((A B)^{2}\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{3 x^{2}-7 x+33}{x^{3}+27} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 1.1 x+0.7 y-3.1 z=-1.79 \\ 2.1 x+0.5 y-1.6 z=-0.13 \\ 0.5 x+0.4 y-0.5 z=-0.07 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 5 x+2 y-z=1 \\ -7 x-8 y+3 z=1.5 \\ 6 x-12 y+z=7 \end{array} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$ \begin{array}{l} \frac{1}{10} x-\frac{1}{5} y+4 z=\frac{-41}{2} \\ \frac{1}{5} x-20 y+\frac{2}{5} z=-101 \\ \frac{3}{10} x+4 y-\frac{3}{10} z=23 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{c} x+2 y-z=1 \\ -x-2 y+2 z=-2 \\ 3 x+6 y-3 z=5 \end{array} $$

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: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \((B A)^{2}\)

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{x^{2}+2 x+40}{x^{3}-125} $$

For the following exercises, graph the inequality. $$ x^{2}+y^{2}<4 $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 0.5 x-0.5 y+0.5 z=10 \\ 0.2 x-0.2 y+0.2 z=4 \\ 0.1 x-0.1 y+0.1 z=2 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 13 x-17 y+16 z=73 \\ -11 x+15 y+17 z=61 \\ 46 x+10 y-30 z=-18 \end{array} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$ \begin{array}{l} \frac{1}{2} x-\frac{1}{5} y+\frac{1}{5} z=\frac{31}{100} \\ -\frac{3}{4} x-\frac{1}{4} y+\frac{1}{2} z=\frac{7}{40} \\ -\frac{4}{5} x-\frac{1}{2} y+\frac{3}{2} z=\frac{1}{4} \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{aligned} x+2 y-z &=1 \\ -x-2 y+2 z &=-2 \\ 3 x+6 y-3 z &=3 \end{aligned} $$

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

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{4 x^{2}+4 x+12}{8 x^{3}-27} $$

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

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 0.1 x+0.2 y+0.3 z=0.37 \\ 0.1 x-0.2 y-0.3 z=-0.27 \\ 0.5 x-0.1 y-0.3 z=-0.03 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} -4 x-3 y-8 z=-7 \\ 2 x-9 y+5 z=0.5 \\ 5 x-6 y-5 z=-2 \end{array} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$ \begin{aligned} &0.1 x+0.2 y+0.3 z=-1.4\\\ &0.1 x-0.2 y+0.3 z=0.6\\\ &0.4 y+0.9 z=-2 \end{aligned} $$

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

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

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{-50 x^{2}+5 x-3}{125 x^{3}-1} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 0.5 x-0.5 y-0.3 z=0.13 \\ 0.4 x-0.1 y-0.3 z=0.11 \\ 0.2 x-0.8 y-0.9 z=-0.32 \end{array} $$

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. $$ \begin{array}{r} -x+2 y=4 \\ 2 x-4 y=1 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{l} 4 x-6 y+8 z=10 \\ -2 x+3 y-4 z=-5 \\ x+y+z=1 \end{array} $$

For the following exercises, use a calculator to solve the system of equations with matrix inverses. $$ \begin{array}{l} 2 x-y=-3 \\ -x+2 y=2.3 \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{c} x+y+z=100 \\ x+2 z=125 \\ -y+2 z=25 \end{array} $$

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

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{-2 x^{3}-30 x^{2}+36 x+216}{x^{4}+216 x} $$

For the following exercises, graph the system of inequalities. Label all points of intersection. $$ \begin{array}{l} x^{2}+y^{2}<25 \\ 3 x^{2}-y^{2}>12 \end{array} $$

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{l} 0.5 x+0.2 y-0.3 z=1 \\ 0.4 x-0.6 y+0.7 z=0.8 \\ 0.3 x-0.1 y-0.9 z=0.6 \end{array} $$

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. $$ \begin{array}{r} x+2 y=7 \\ 2 x+6 y=12 \end{array} $$

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 4 x-6 y+8 z=10 \\ -2 x+3 y-4 z=-5 \\ 12 x+18 y-24 z=-30 \end{array} $$

For the following exercises, use a calculator to solve the system of equations with matrix inverses. $$ \begin{array}{l} -\frac{1}{2} x-\frac{3}{2} y=-\frac{43}{20} \\ \frac{5}{2} x+\frac{11}{5} y=\frac{31}{4} \end{array} $$

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{l} \frac{1}{4} x-\frac{2}{3} z=-\frac{1}{2} \\ \frac{1}{5} x+\frac{1}{3} y=\frac{4}{7} \\ \frac{1}{5} y-\frac{1}{3} z=\frac{2}{9} \end{array} $$

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

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$ \frac{3 x^{3}+2 x^{2}+14 x+15}{\left(x^{2}+4\right)^{2}} $$

For the following exercises, graph the system of inequalities. Label all points of intersection. $$ \begin{array}{l} x^{2}-y^{2}>-4 \\ x^{2}+y^{2}<12 \end{array} $$