Chapter 11

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

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

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+y+z &=0 \\\\-x+2 y+5 z &=3 \\ 3 x-y &=6 \end{aligned}\right. $$

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

\(15-20\) 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. $$ \left\\{\begin{array}{c}{12 x+15 y=-18} \\ {2 x+\frac{5}{2} y=-3}\end{array}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don’t calculate the inverse. $$ \left[\begin{array}{rrr}{30} & {0} & {20} \\ {0} & {-10} & {-20} \\ {40} & {0} & {10}\end{array}\right] $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x+y & \leq 4 \\ y & \geq x \end{aligned}\right. $$

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

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} x+y+z &=2 \\ 2 x-3 y+2 z &=4 \\ 4 x+y-3 z &=1 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x+14}{x^{2}-2 x-8} $$

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

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right] $$ $$ D=\left[\begin{array}{ll}{7} & {3}\end{array}\right] \quad E=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right] \quad F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right] \quad H=\left[\begin{array}{rr}{3} & {1} \\ {2} & {-1}\end{array}\right] $$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ \begin{array}{ll}{\text { (a) } B+C} & {\text { (b) } B+F}\end{array} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x &-4 z=1 \\ 2 x-y-6 z &=4 \\ 2 x+3 y-2 z &=8 \end{aligned}\right. $$

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

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

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} x+y+z &=4 \\\\-x+2 y+3 z &=17 \\ 2 x-y &=-7 \end{aligned}\right. $$

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{array}{l}{x-y=3} \\ {x+3 y=7}\end{array}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{8 x-3}{2 x^{2}-x} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rrr}{3} & {-2} & {0} \\ {5} & {1} & {1} \\ {2} & {-2} & {0}\end{array}\right] $$

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right] $$ $$ D=\left[\begin{array}{ll}{7} & {3}\end{array}\right] \quad E=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right] \quad F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right] \quad H=\left[\begin{array}{rr}{3} & {1} \\ {2} & {-1}\end{array}\right] $$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ \begin{array}{ll}{\text { (a) } C-B} & {\text { (b) } 2 C-6 B}\end{array} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-y+2 z &=2 \\ 3 x+y+5 z &=8 \\ 2 x-y-2 z &=-7 \end{aligned}\right. $$$17-36$ . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-y+2 z &=2 \\ 3 x+y+5 z &=8 \\ 2 x-y-2 z &=-7 \end{aligned}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don’t calculate the inverse. $$ \left[\begin{array}{lll}{1} & {3} & {7} \\ {2} & {0} & {8} \\ {0} & {2} & {2}\end{array}\right] $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{l}{y<\frac{1}{4} x+2} \\ {y \geq 2 x-5}\end{array}\right. $$

$$ \left\\{\begin{aligned} x-y &=4 \\ x y &=12 \end{aligned}\right. $$

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} x+2 y-z &=-2 \\ x &+z= 0 \\ 2 x-y-z &=-3 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x}{8 x^{2}-10 x+3} $$

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

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right] $$ $$ D=\left[\begin{array}{ll}{7} & {3}\end{array}\right] \quad E=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right] \quad F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right] \quad H=\left[\begin{array}{rr}{3} & {1} \\ {2} & {-1}\end{array}\right] $$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ \begin{array}{ll}{\text { (a) } 5 A} & {\text { (b) } C-5 A}\end{array} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} 2 x+4 y-z &=2 \\ x+2 y-3 z &=-4 \\ 3 x-y+z &=1 \end{aligned}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don’t calculate the inverse. $$ \left[\begin{array}{rrr}{0} & {-1} & {0} \\ {2} & {6} & {4} \\ {1} & {0} & {3}\end{array}\right] $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{l}{x-y>0} \\ {4+y \leq 2 x}\end{array}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{aligned} x y &=24 \\ 2 x^{2}-y^{2}+4 &=0 \end{aligned}\right. $$

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 2 y+z &=4 \\ x+y &=4 \\ 3 x+3 y-z &=10 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{7 x-3}{x^{3}+2 x^{2}-3 x} $$

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{array}{c}{3 x+2 y=0} \\ {-x-2 y=8}\end{array}\right. $$

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

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right] $$ $$ D=\left[\begin{array}{ll}{7} & {3}\end{array}\right] \quad E=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right] \quad F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right] \quad H=\left[\begin{array}{rr}{3} & {1} \\ {2} & {-1}\end{array}\right] $$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ \begin{array}{ll}{\text { (a) } 3 B+2 C} & {\text { (b) } 2 H+D}\end{array} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{array}{rr}{2 x+y-z=} & {-8} \\ {-x+y+z=} & {3} \\ {-2 x} & {+4 z=18}\end{array}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don’t calculate the inverse. $$ \left[\begin{array}{rrrr}{1} & {3} & {3} & {0} \\ {0} & {2} & {0} & {1} \\\ {-1} & {0} & {0} & {2} \\ {1} & {6} & {4} & {1}\end{array}\right] $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{l}{y \leq-2 x+8} \\ {y \leq-\frac{1}{2} x+5} \\ {x \geq 0, \quad y \geq 0}\end{array}\right. $$

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} x_{1}+2 x_{2}-x_{3} &=9 \\ 2 x_{1} -x_{3}=-2 \\ 3 x_{1}+5 x_{2}+2 x_{3} &=22 \end{aligned}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{aligned} x^{2} y &=16 \\ x^{2}+4 y+16 &=0 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{9 x^{2}-9 x+6}{2 x^{3}-x^{2}-8 x+4} $$

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right] $$ $$ D=\left[\begin{array}{ll}{7} & {3}\end{array}\right] \quad E=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right] \quad F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right] \quad H=\left[\begin{array}{rr}{3} & {1} \\ {2} & {-1}\end{array}\right] $$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ \begin{array}{ll}{\text { (a) } A D} & {\text { (b) } D A}\end{array} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} y-2 z &=0 \\ 2 x+3 y &=2 \\\\-x-2 y+z &=-1 \end{aligned}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don’t calculate the inverse. $$ \left[\begin{array}{rrrr}{1} & {2} & {0} & {2} \\ {3} & {-4} & {0} & {4} \\\ {0} & {1} & {6} & {0} \\ {1} & {0} & {2} & {0}\end{array}\right] $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{c}{4 x+3 y \leq 18} \\ {2 x+y \leq 8} \\ {x \geq 0, \quad y \geq 0}\end{array}\right. $$

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{array}{ll}{2 x_{1}+x_{2}} & {=7} \\ {2 x_{1}-x_{2}+x_{3}} & {=6} \\ {3 x_{1}-2 x_{2}+4 x_{3}} & {=11}\end{array}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{x+\sqrt{y}=0} \\ {y^{2}-4 x^{2}=12}\end{array}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{9 x^{2}-9 x+6}{2 x^{3}-x^{2}-8 x+4} $$