Chapter 11

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{2 x}{(x-1)(x+1)} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{cc}{\frac{1}{2}} & {\frac{1}{3}} \\ {5} & {4}\end{array}\right] $$

\(13-16\) . Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \begin{array}{l}{\left\\{\begin{aligned} x+y-3 z &=3 \\\\-2 x+3 y+z &=2 \\\ x-y+2 z &=0 \end{aligned}\right.} \\ {\text { Eliminate the } x \text { -term }} \\ {\text { from the second equation. }}\end{array} $$

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{2} & {-3} \\ {0} & {1} \\ {1} & {2}\end{array}\right]\left[\begin{array}{l}{5} \\ {1}\end{array}\right] $$

\(13-14\) . Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$ \left\\{\begin{array}{r}{x+y=2} \\ {2 x+y=5}\end{array}\right. $$

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{12}, A_{12} $$

\(3-16=\) Graph the inequality. $$ x^{2}+y^{2} \leq 25 $$

\(15-18\) . Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$ \left\\{\begin{array}{l}{x^{2}+y=8} \\ {x-2 y=-6}\end{array}\right. $$

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$ \left[\begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\\ {0} & {1} & {5} & {1}\end{array}\right] $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{5}{(x-1)(x+4)} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rr}{0.4} & {-1.2} \\ {0.3} & {0.6}\end{array}\right] $$

Solve the matrix equation for the unknown matrix \(X,\) or explain why no solution exists. $$ \begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right] \quad D=\left[\begin{array}{cc}{10} & {20} \\\ {30} & {20} \\ {10} & {0}\end{array}\right]}\end{array} $$ $$ 2 X+A=B $$

\(13-16\) . Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \left\\{\begin{array}{c}{2 x-y+3 z=2} \\ {x+2 y-z=4} \\ {-4 x+5 y+z=10} \\\ {\text { Eliminate the } x \text { -term }} \\ {\text { from the third equation. }}\end{array}\right. $$

\(15-20\) m Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$ \left\\{\begin{array}{r}{x-y=4} \\ {2 x+y=2}\end{array}\right. $$

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{13}, A_{13} $$

\(3-16=\) Graph the inequality. $$ x^{2}+(y-1)^{2} \leq 1 $$

\(15-18\) . Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$ \left\\{\begin{array}{l}{x-y^{2}=-4} \\ {x-y=2}\end{array}\right. $$

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$ \left[\begin{array}{llll}{1} & {0} & {0} & {1} \\ {0} & {1} & {0} & {2} \\\ {0} & {0} & {1} & {3}\end{array}\right] $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x+6}{x(x+3)} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{lll}{4} & {2} & {3} \\ {3} & {3} & {2} \\ {1} & {0} & {1}\end{array}\right] $$

\(13-16\) . Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \begin{array}{c}{x-4 y+z=3} \\ {y-3 z=10} \\ {3 y-8 z=24} \\ {\text { Eliminate the } y \text { -term }} \\ {\text { from the third equation. }}\end{array} $$

\(15-20\) m Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$ \left\\{\begin{array}{l}{2 x-y=4} \\ {3 x+y=6}\end{array}\right. $$

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{23}, A_{23} $$

\(15-18\) . Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$ \left\\{\begin{aligned} x^{2}+y &=0 \\ x^{3}-2 x-y &=0 \end{aligned}\right. $$

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$ \left[\begin{array}{rrrrr}{1} & {3} & {0} & {-1} & {0} \\ {0} & {0} & {1} & {2} & {0} \\ {0} & {0} & {0} & {0} & {1} \\ {0} & {0} & {0} & {0} & {0}\end{array}\right] $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{12}{x^{2}-9} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rrr}{2} & {4} & {1} \\ {-1} & {1} & {-1} \\ {1} & {4} & {0}\end{array}\right] $$

Solve the matrix equation for the unknown matrix \(X,\) or explain why no solution exists. $$ \begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right] \quad D=\left[\begin{array}{cc}{10} & {20} \\\ {30} & {20} \\ {10} & {0}\end{array}\right]}\end{array} $$ $$ 2(B-X)=D $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{array}{rr}{x-y-z=} & {4} \\ {2 y+z=} & {-1} \\ {-x+y-2 z=} & {5}\end{array}\right. $$

\(15-20\) m Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$ \left\\{\begin{array}{l}{2 x-3 y=12} \\ {-x+\frac{3}{2} y=4}\end{array}\right. $$

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{32}, A_{32} $$

\(15-18\) . Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=4 x \\ x &=y^{2} \end{aligned}\right. $$

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$ \left[\begin{array}{cccccc}{1} & {3} & {0} & {1} & {0} & {0} \\ {0} & {1} & {0} & {4} & {0} & {0} \\ {0} & {0} & {0} & {1} & {1} & {2} \\ {0} & {0} & {0} & {1} & {0} & {0}\end{array}\right] $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x-12}{x^{2}-4 x} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rrr}{5} & {7} & {4} \\ {3} & {-1} & {3} \\ {6} & {7} & {5}\end{array}\right] $$

Solve the matrix equation for the unknown matrix \(X,\) or explain why no solution exists. $$ \begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right] \quad D=\left[\begin{array}{cc}{10} & {20} \\\ {30} & {20} \\ {10} & {0}\end{array}\right]}\end{array} $$ $$ 5(X-C)=D $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-y+z &=0 \\ y+2 z &=-2 \\ x+y-z &=2 \end{aligned}\right. $$

\(15-20\) m Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$ \left\\{\begin{aligned} 2 x+6 y &=0 \\\\-3 x-9 y &=18 \end{aligned}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don’t calculate the inverse. $$ \left[\begin{array}{rrr}{2} & {1} & {0} \\ {0} & {-2} & {4} \\ {0} & {1} & {-3}\end{array}\right] $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{y+x^{2}=4 x} \\ {y+4 x=16}\end{array}\right. $$

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} x-2 y+z &=1 \\ y+2 z &=5 \\ x+y+3 z &=8 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{4}{x^{2}-4} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rrr}{1} & {2} & {3} \\ {4} & {5} & {-1} \\ {1} & {-1} & {-10}\end{array}\right] $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+y+z &=4 \\ x+3 y+3 z &=10 \\ 2 x+y-z &=3 \end{aligned}\right. $$

Solve the matrix equation for the unknown matrix \(X,\) or explain why no solution exists. $$ \begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right] \quad D=\left[\begin{array}{cc}{10} & {20} \\\ {30} & {20} \\ {10} & {0}\end{array}\right]}\end{array} $$ $$ \frac{1}{5}(X+D)=C $$

\(15-20\) m Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$ \left\\{\begin{array}{l}{-x+\frac{1}{2} y=-5} \\ {2 x-y=10}\end{array}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don’t calculate the inverse. $$ \left[\begin{array}{rrr}{1} & {2} & {5} \\ {-2} & {-3} & {2} \\ {3} & {5} & {3}\end{array}\right] $$

\(17-20\) . An equation and its graph are given. Find an inequality whose solution is the shaded region. $$ y=x^{3}-4 x $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{x-y^{2}=0} \\ {y-x^{2}=0}\end{array}\right. $$

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l}{x+y+6 z=3} \\ {x+y+3 z=3} \\ {x+2 y+4 z=7}\end{array}\right. $$