Chapter 11

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{aligned} x-y+3 z &=3 \\ 4 x-8 y+32 z &=24 \\ 2 x-3 y+11 z &=4 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{-10 x^{2}+27 x-14}{(x-1)^{3}(x+2)} $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{r}{3 x+2 y=8} \\ {x-2 y=0}\end{array}\right. $$

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

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+3 y-2 z &=0 \\ 2 x+& 4 z=4 \\ 4 x+6 y &=4 \end{aligned}\right. $$

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{array}{l}{6 x+12 y=33} \\ {4 x+7 y=20}\end{array}\right. $$

$$ \left\\{\begin{aligned} x &>0 \\ y &>0 \\ x+y &<10 \\ x^{2}+y^{2} &>9 \end{aligned}\right. $$

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{array}{rr}{-2 x+6 y-2 z=} & {-12} \\ {x-3 y+2 z=} & {10} \\\ {-x+3 y+2 z=} & {6}\end{array}\right. $$

\(33-40=\) Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. $$ \left\\{\begin{array}{l}{y=x^{2}-4 x} \\ {2 x-y=2}\end{array}\right. $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{aligned} 4 x+2 y &=16 \\ x-5 y &=70 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{-2 x^{2}+5 x-1}{x^{4}-2 x^{3}+2 x-1} $$

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. $$ \begin{array}{ll}{\text { (a) } D B+D C} & {\text { (b) } B F+F E}\end{array} $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} 2 x+4 y-z &=3 \\ x+2 y+4 z &=6 \\ x+2 y-2 z &=0 \end{aligned}\right. $$

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned} x-6 y &=3 \\ 3 x+2 y &=1 \end{aligned}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{c}{x^{2}-y \leq 0} \\ {2 x^{2}+y \leq 12}\end{array}\right. $$

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{aligned} x+4 y-2 z &=-3 \\ 2 x-y+5 z &=12 \\ 8 x+5 y+11 z &=30 \end{aligned}\right. $$

\(33-40=\) Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. $$ \begin{array}{l}{x^{2}+y^{2}=25} \\ {x+3 y=2}\end{array} $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{aligned} x+4 y &=8 \\ 3 x+12 y &=2 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{3 x^{3}+22 x^{2}+53 x+41}{(x+2)^{2}(x+3)^{2}} $$

Solve for \(x\) and \(y\) $$ \left[\begin{array}{cc}{x} & {2 y} \\ {4} & {6}\end{array}\right]=\left[\begin{array}{cc}{2} & {-2} \\ {2 x} & {-6 y}\end{array}\right] $$

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

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{array}{l}{\frac{1}{2} X+\frac{1}{3} y=1} \\ {\frac{1}{4} x-\frac{1}{6} y=-\frac{3}{2}}\end{array}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{l}{x^{2}+y^{2}<9} \\ {2 x+y^{2} \geq 1}\end{array}\right. $$

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{aligned} 3 r+2 s-3 t &=10 \\ r-s-t &=-5 \\ r+4 s-t &=20 \end{aligned}\right. $$

\(33-40=\) Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=17 \\ x^{2}-2 x+y^{2} &=13 \end{aligned}\right. $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{aligned}-3 x+5 y &=2 \\ 9 x-15 y &=6 \end{aligned}\right. $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{3 x^{2}+12 x-20}{x^{4}-8 x^{2}+16} $$

Solve for \(x\) and \(y\) $$ 3\left[\begin{array}{ll}{x} & {y} \\ {y} & {x}\end{array}\right]=\left[\begin{array}{rr}{6} & {-9} \\ {-9} & {6}\end{array}\right] $$

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+y+z+w &=0 \\ x+y+2 z+2 w &=0 \\ 2 x+2 y+3 z+4 w &=1 \\ 2 x+3 y+4 z+5 w &=2 \end{aligned}\right. $$

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{array}{l}{0.4 x+1.2 y=0.4} \\ {1.2 x+1.6 y=3.2}\end{array}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{c}{x+2 y \leq 14} \\ {3 x-y \geq 0} \\ {x-y \geq 2}\end{array}\right. $$

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{array}{l}{2 x+y-2 z=12} \\ {-x-\frac{1}{2} y+z=-6} \\ {3 x+\frac{3}{2} y-3 z=18}\end{array}\right. $$

\(33-40=\) Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. $$ \left\\{\begin{array}{l}{\frac{x^{2}}{9}+\frac{y^{2}}{18}=1} \\ {y=-x^{2}+6 x-2}\end{array}\right. $$

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

\(37-38\) . Finance An investor has \(\$ 100,000\) to invest in three types of bonds: short-term, intermediate-term, and long-term. How much should she invest in each type to satisfy the given conditions? Short-term bonds pay 4\(\%\) annually, intermediate-term bonds pay \(5 \%,\) and long-term bonds pay \(6 \% .\) The investor wishes to realize a total annual income of 5.1\(\%\) , with equal amounts invested in short- and intermediate- term bonds.

Solve for \(x\) and \(y\) $$ 2\left[\begin{array}{cc}{x} & {y} \\ {x+y} & {x-y}\end{array}\right]=\left[\begin{array}{rr}{2} & {-4} \\ {-2} & {6}\end{array}\right] $$

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{array}{l}{10 x-17 y=21} \\ {20 x-31 y=39}\end{array}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{c}{y < x+6} \\ {3 x+2 y \geq 12} \\ {x-2 y \leq 2}\end{array}\right. $$

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{aligned} y-5 z &=7 \\ 3 x+2 y &=12 \\ 3 x &+10 z=80 \end{aligned}\right. $$

Use a calculator that can perform matrix operations to solve the system, as in Example 7. $$ \left\\{\begin{array}{l}{x+y+z+w=15} \\ {x-y+z-w=5} \\ {x+2 y+3 z+4 w=26} \\\ {x-2 y+3 z-4 w=2}\end{array}\right. $$

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{c}{2 x-3 y=-8} \\ {14 x-21 y=3}\end{array}\right. $$

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

Solve for \(x\) and \(y\) $$ \left[\begin{array}{rr}{x} & {y} \\ {-y} & {x}\end{array}\right]-\left[\begin{array}{rr}{y} & {x} \\ {x} & {-y}\end{array}\right]=\left[\begin{array}{rr}{4} & {-4} \\ {-6} & {6}\end{array}\right] $$

\(37-38\) . Finance An investor has \(\$ 100,000\) to invest in three types of bonds: short-term, intermediate-term, and long-term. How much should she invest in each type to satisfy the given conditions? Short-term bonds pay 4\(\%\) annually, intermediate-term bonds pay \(6 \%,\) and long-term bonds pay \(8 \% .\) The investor wishes to have a total annul return of \(\$ 6700\) on her investment, with equal amounts invested in intermediate- and long-term bonds.

Use Cramer’s Rule to solve the system. $$ \left\\{\begin{aligned} x-y+2 z =0 \\ 3 x \quad\quad+z=11 \\\\-x+2 y\quad\quad =0 \end{aligned}\right. $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x & \leq 5 \\ x+y & \leq 7 \end{aligned}\right. $$

Solve the system of linear equations. $$ \left\\{\begin{aligned} 4 x-3 y+z &=-8 \\\\-2 x+y-3 z &=-4 \\ x-y+2 z &=3 \end{aligned}\right. $$

\(33-40=\) Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. $$ \left\\{\begin{array}{l}{x^{4}+16 y^{4}=32} \\ {x^{2}+2 x+y=0}\end{array}\right. $$

Solve the matrix equation by multiplying each side by the appropriate inverse matrix. $$ \left[\begin{array}{rr}{3} & {-2} \\ {-4} & {3}\end{array}\right]\left[\begin{array}{lll}{x} & {y} & {z} \\ {u} & {v} & {w}\end{array}\right]=\left[\begin{array}{rrr}{1} & {0} & {-1} \\ {2} & {1} & {3}\end{array}\right] $$

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{2 x^{3}+7 x+5}{\left(x^{2}+x+2\right)\left(x^{2}+1\right)} $$