Chapter 11

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{r}{x+y=7} \\ {2 x-3 y=-1}\end{array}\right. $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} 2 y+z &=3 \\ 5 x+4 y+3 z &=-1 \\ x-3 y &=-2 \end{aligned}\right. $$

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. $$ \begin{array}{ll}{\text { (a) } D H} & {\text { (b) } H D}\end{array} $$

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{llll}{0} & {0} & {4} & {6} \\ {2} & {1} & {1} & {3} \\\ {2} & {1} & {2} & {3} \\ {3} & {0} & {1} & {7}\end{array}\right| $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \\ 3 x+2 y & \leq 9 \end{aligned}\right. $$

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l}{2 x-3 y-z=13} \\ {-x+2 y-5 z=6} \\ {5 x-y-z=49}\end{array}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{x^{2}+y^{2}=9} \\ {x^{2}-y^{2}=1}\end{array}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x^{2}+1}{x^{3}+x^{2}} $$

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

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. $$ \left(\begin{array}{llll}{\text { a) }} & {A H} & {} & {\text { (b) }} & {H A}\end{array}\right. $$

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{rrrr}{-2} & {3} & {-1} & {7} \\ {4} & {6} & {-2} & {3} \\\ {7} & {7} & {0} & {5} \\ {3} & {-12} & {4} & {0}\end{array}\right| $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x &>2 \\ y &<12 \\ 2 x-4 y &>8 \end{aligned}\right. $$

The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l}{10 x+10 y-20 z=60} \\ {15 x+20 y+30 z=-25} \\ {-5 x+30 y-10 z=45}\end{array}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{aligned} x^{2}+2 y^{2} &=2 \\ 2 x^{2}-3 y &=15 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{3 x^{2}+5 x-13}{(3 x+2)\left(x^{2}-4 x+4\right)} $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{l}{4 x-3 y=28} \\ {9 x-y=-6}\end{array}\right. $$

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. $$ \left\\{\begin{aligned} x+2 y-z &=1 \\ 2 x+3 y-4 z &=-3 \\ 3 x+6 y-3 z &=4 \end{aligned}\right. $$

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{ccccc}{1} & {2} & {3} & {4} & {5} \\ {0} & {2} & {4} & {6} & {8} \\ {0} & {0} & {3} & {6} & {9} \\ {0} & {0} & {0} & {4} & {8} \\\ {0} & {0} & {0} & {0} & {5}\end{array}\right| $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{l}{y \leq 9-x^{2}} \\ {x \geq 0, \quad y \geq 0}\end{array}\right. $$

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{aligned} x+y+z &=2 \\ y-3 z &=1 \\ 2 x+y+5 z &=0 \end{aligned}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{aligned} 2 x^{2}-8 y^{3} &=19 \\ 4 x^{2}+16 y^{3} &=34 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{2 x}{4 x^{2}+12 x+9} $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{l}{x+2 y=7} \\ {5 x-y=2}\end{array}\right. $$

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. $$ \begin{array}{ll}{\text { (a) } G F} & {\text { (b) } G E}\end{array} $$

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

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{rrrr}{2} & {-1} & {6} & {4} \\ {7} & {2} & {-2} & {5} \\\ {4} & {-2} & {10} & {8} \\ {6} & {1} & {1} & {4}\end{array}\right| $$

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

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{array}{r}{x+3 z=3} \\ {2 x+y-2 z=5} \\ {-y+8 z=8}\end{array}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{aligned} x^{4}+y^{3} &=17 \\ 3 x^{4}+5 y^{3} &=53 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{x-4}{(2 x-5)^{2}} $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{l}{-4 x+12 y=0} \\ {12 x+4 y=160}\end{array}\right. $$

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. $$ \begin{array}{ll}{\text { (a) } B^{2}} & {\text { (b) } F^{2}}\end{array} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-2 y-3 z &=5 \\ 2 x+y-z &=5 \\ 4 x-3 y-7 z &=5 \end{aligned}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{l}{y<9-x^{2}} \\ {y \geq x+3}\end{array}\right. $$

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{aligned} 2 x-3 y-9 z &=-5 \\ x +3 z=2\\\\-3 x+y-4 z &=-3 \end{aligned}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{array}{c}{\frac{2}{x}-\frac{3}{y}=1} \\\ {-\frac{4}{x}+\frac{7}{y}=1}\end{array}\right. $$

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

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{l}{\frac{1}{2} x+\frac{1}{3} y=2} \\ {\frac{1}{5} x-\frac{2}{3} y=8}\end{array}\right. $$

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. $$ A^{2} \quad \text { (b) } A^{3} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+y-z &=0 \\ x+2 y-3 z &=-3 \\ 2 x+3 y-4 z &=-3 \end{aligned}\right. $$

Consider the system $$ \left\\{\begin{aligned} x+2 y+6 z &=5 \\\\-3 x-6 y+5 z &=8 \\ 2 x+6 y+9 z &=7 \end{aligned}\right. $$ (a) Verify that \(x=-1, y=0, z=1\) is a solution of the system. (b) Find the determinant of the coefficient matrix. (c) Without solving the system, determine whether there are any other solutions. (d) Can Cramer's Rule be used to solve this system? Why or why not?

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} y & \geq x^{2} \\ x+y & \geq 6 \end{aligned}\right. $$

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{array}{rr}{x-2 y+5 z=} & {3} \\ {-2 x+6 y-11 z=} & {1} \\ {3 x-16 y-20 z=} & {-26}\end{array}\right. $$

\(19-32\) . Find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{\frac{4}{x^{2}}+\frac{6}{y^{4}}=\frac{7}{2}} \\\ {\frac{1}{x^{2}}-\frac{2}{y^{4}}=0}\end{array}\right. $$

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

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. $$ \begin{array}{ll}{\text { (a) }(D A) B} & {\text { (b) } D(A B)}\end{array} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-2 y+z &=3 \\ 2 x-5 y+6 z &=7 \\ 2 x-3 y-2 z &=5 \end{aligned}\right. $$

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned} 2 x-y &=-9 \\ x+2 y &=8 \end{aligned}\right. $$

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