Chapter 10

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{aligned} 10 x+10 y-20 z &=60 \\ 15 x+20 y+30 z &=-25 \\\\-5 x+30 y-10 z &=45 \end{aligned}\right.$$

Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned}-x+2 y+5 z &=4 \\ x-& 2 z=0 \\ 4 x-2 y-11 z &=2 \end{aligned}\right. $$

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l}{y<9-x^{2}} \\ {y \geq x+3}\end{array}\right.$$

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

23-26 m Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{lllll}{1} & {2} & {3} & {4} & {5} \\ {0} & {2} & {4} & {6} & {8} \\ {0} & {0} & {3} & {6} & {9} \\ {0} & {0} & {0} & {4} & {8} \\\ {0} & {0} & {0} & {0} & {5}\end{array}\right| $$

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. $$ A D $$

Solve the system of equations by converting to a matrix equation and using the inverse of the coefficient matrix, as in Example 6. Use the inverses from Exercises 7–10, 15, 16, 19, and 21. \(\left\\{\begin{aligned} 2 x+5 y &=2 \\\\-5 x-13 y &=20 \end{aligned}\right.\)

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. $$\left\\{\begin{array}{l}{6 x+4 y=12} \\ {9 x+6 y=18}\end{array}\right.$$

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} x+y+z =2 \\ y-3 z &=1 \\ 2 x+y+5 z =0 \end{aligned}\right.$$

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

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{c}\qquad {y \geq x^{2}} \\ {x+y \geq 6}\end{array}\right.$$

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

23-26 m Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{rrrr}{2} & {-1} & {6} & {4} \\ {7} & {2} & {-2} & {5} \\\ {4} & {-2} & {10} & {8} \\ {6} & {1} & {1} & {4}\end{array}\right| $$

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. $$ B C $$

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. $$\left\\{\begin{aligned} 25 x-75 y &=100 \\\\-10 x+30 y &=-40 \end{aligned}\right.$$

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

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

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

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

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

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. $$ B F $$

Solve the system of equations by converting to a matrix equation and using the inverse of the coefficient matrix, as in Example 6. Use the inverses from Exercises 7–10, 15, 16, 19, and 21. \(\left\\{\begin{aligned} 2 x+4 y+z &=7 \\\\-x+y-z &=0 \\ x+4 y &=-2 \end{aligned}\right.\)

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. $$\left\\{\begin{array}{l}{8 s-3 t=-3} \\ {5 s-2 t=-1}\end{array}\right.$$

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

Find all solutions of the system of equations. \(\left\\{\begin{aligned} x-y &=4 \\ x y &=12 \end{aligned}\right.\)

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x &>0 \\ y &>0 \\ x+y &<10 \\ x^{2}+y^{2} &>9 \end{aligned}\right.$$

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

Consider the system $$ \left\\{\begin{array}{c}{x+2 y+6 z=5} \\ {-3 x-6 y+5 z=8} \\ {2 x+6 y+5 z=7} \\\ {\text { (a) Verify that } x=-1, y=0, z=7} \\ {\text { system. }} \\\ {\text { (b) Find the determinant of the coefficient matrix. }}\end{array}\right. $$ (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?

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. $$ G F $$

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. $$\left\\{\begin{aligned} u-30 v &=-5 \\\\-3 u+80 v &=5 \end{aligned}\right.$$

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{array}{rr}{x-2 y+5 z=} & {3} \\ {-2 x+6 y-11 z=} & {1} \\\ {3 x-16 y+20 z=} & {-26}\end{array}\right.$$

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

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

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

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

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{c}{2 x-y=-9} \\ {x+2 y=8}\end{array}\right. $$

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. $$ (D A) B $$

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. $$\left\\{\begin{array}{l}{\frac{1}{2} x+\frac{3}{5} y=3} \\ {\frac{5}{3} x+2 y=10}\end{array}\right.$$

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{aligned} x-y+3 z &=3 \\ 4 x-8 y+32 z &=24 \\ 2 x-3 y+11 z &=4 \end{aligned}\right.$$

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

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l}{x^{2}+y^{2}<9} \\ {2 x+y^{2} \geq 1}\end{array}\right.$$

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

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{l}{6 x+12 y=33} \\ {4 x+7 y=20}\end{array}\right. $$

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. $$ D(A B) $$

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. $$\left\\{\begin{array}{l}{\frac{3}{2} x-\frac{1}{3} y=\frac{1}{2}} \\ {2 x-\frac{1}{2} y=-\frac{1}{2}}\end{array}\right.$$

25–34 Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{array}{rr}{-2 x+6 y-2 z=} & {-12} \\ {x-3 y+2 z=} & {10} \\\ {-x+3 y+2 z=} & {6}\end{array}\right.$$

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