Chapter 10

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x+2 y & \leq 14 \\ 3 x-y & \geq 0 \\ x-y & \geq 2 \end{aligned}\right.$$

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

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{c}{x-6 y=3} \\ {3 x+2 y=1}\end{array}\right. $$

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

Use a calculator that can perform matrix operations to solve the system, as in Example 7. \(\left\\{\begin{aligned} x+y-2 z &=3 \\ 2 x+& 5 z=11 \\ 2 x+3 y &=12 \end{aligned}\right.\)

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 3. $$\left\\{\begin{aligned} 0.4 x+1.2 y &=14 \\ 12 x-5 y &=10 \end{aligned}\right.$$

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} x+4 y-2 z &=-3 \\ 2 x-y+5 z &=12 \\ 8 x+5 y+11 z &=30 \end{aligned}\right.$$

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

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} y & < x+6 \\ 3 x+2 y & \geq 12 \\ x-2 y & \leq 2 \end{aligned}\right.$$

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

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{l}{\frac{1}{2} x+\frac{1}{3} y=1} \\ {\frac{1}{4} x-\frac{1}{6} y=-\frac{3}{2}}\end{array}\right. $$

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

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

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 3. $$\left\\{\begin{aligned} 26 x-10 y &=-4 \\\\-0.6 x+1.2 y &=3 \end{aligned}\right.$$

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} 3 r+2 s-3 t &=10 \\ r-s-t &=-5 \\ r+4 s-t &=20 \end{aligned}\right.$$

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

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x & \leq 5 \\ x+y & \leq 7 \end{aligned}\right.$$

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

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{l}{0.4 x+1.2 y=0.4} \\ {1.2 x+1.6 y=3.2}\end{array}\right. $$

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

Use a calculator that can perform matrix operations to solve the system, as in Example 7. \(\left\\{\begin{array}{l}{12 x+\frac{1}{2} y-7 z=21} \\ {11 x-2 y+3 z=43} \\\ {13 x+y-4 z=29}\end{array}\right.\)

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 3. $$\left\\{\begin{aligned} \frac{1}{3} x-\frac{1}{4} y &=2 \\\\-8 x+6 y &=10 \end{aligned}\right.$$

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{array}{l}{2 x+y-2 z=12} \\ {-x-\frac{1}{2} y+z=-6} \\ {3 x+\frac{3}{2} y-3 z=18}\end{array}\right.$$

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? $$ \begin{array}{l}{\text { Short-term bonds pay } 4 \% \text { annually, intermediate-term bonds }} \\ {\text { pay } 5 \% \text { , and long-term bonds pay } 6 \% . \text { The investor wishes }} \\ {\text { to realize a total annual income of } 5.1 \%, \text { with equal }} \\ {\text { amounts invested in short- and intermediate-term bonds. }}\end{array} $$

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ y & \leq 4 \\ 2 x+y & \leq 8 \end{aligned}\right.$$

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

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{l}{10 x-17 y=21} \\ {20 x-31 y=39}\end{array}\right. $$

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

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 3. $$\left\\{\begin{aligned}-\frac{1}{10} x+\frac{1}{2} y &=4 \\ 2 x-10 y &=-80 \end{aligned}\right.$$

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} y-5 z \qquad=7 \\ 3 x+2 y \qquad=12 \\ 3 x \qquad +10 z=80 \end{aligned}\right.$$

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? $$ \begin{array}{l}{\text { Short-term bonds pay } 4 \% \text { annually, intermediate-term bonds }} \\ {\text { pay } 6 \%, \text { and long-term bonds pay } 8 \% . \text { The investor wishes }} \\ {\text { to have a total annual return of } \$ 6700 \text { on her investment, }} \\ {\text { with equal amounts invested in intermediate- and long-term }} \\ {\text { bonds. }}\end{array} $$

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} y &>x+1 \\ x+2 y & \leq 12 \\ x+1 &>0 \end{aligned}\right.$$

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

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{aligned} x-y+2 z &=0 \\ 3 x &+z=11 \\\\-x+2 y &=0 \end{aligned}\right. $$

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

Use a calculator that can perform matrix operations to solve the system, as in Example 7. \(\left\\{\begin{array}{rr}{x+y} & {-3 w=} & {0} \\ {x} & {-2 z=} & {8} \\ {2 y-z+w} & {=5} \\ {2 x+3 y} & {-2 w=} & {13}\end{array}\right.\)

Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system correct to two decimal places, either by zooming in and using TRACE or by using Intersect. $$\left\\{\begin{array}{l}{0.21 x+3.17 y=9.51} \\ {2.35 x-1.17 y=5.89}\end{array}\right.$$

35–46 Solve the system of linear equations. $$\left\\{\begin{aligned} 4 x-3 y+z &=-8 \\\\-2 x+y-3 z &=-4 \\ x-y+2 z &=3 \end{aligned}\right.$$

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x+y &>12 \\ y &<\frac{1}{2} x-6 \\ 3 x+y &<6 \end{aligned}\right.$$

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

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{aligned} 5 x-3 y+z &=6 \\ 4 y-6 z &=22 \\ 7 x+10 y &=-13 \end{aligned}\right. $$

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

Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system correct to two decimal places, either by zooming in and using TRACE or by using Intersect. $$\left\\{\begin{aligned} 18.72 x-14.91 y &=12.33 \\ 6.21 x-12.92 y &=17.82 \end{aligned}\right.$$

35–46 Solve the system of linear equations. $$\left\\{\begin{array}{l}{2 x-3 y+5 z=14} \\ {4 x-y-2 z=-17} \\\ {-x-y+z=3}\end{array}\right.$$