Chapter 10

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. \(\left\\{\begin{array}{c}{x+y=2} \\ {2 x+y=5}\end{array}\right.\)

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

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

\(15-22\) . Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$ \left[\begin{array}{rrr}{30} & {0} & {20} \\ {0} & {-10} & {-20} \\ {40} & {0} & {10}\end{array}\right] $$

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. $$ C-B $$

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

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{aligned} x+2 y-z =-2 \\ x \qquad+z=0 \\ 2 x-y-z =-3 \end{aligned}\right.$$

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. $$\left\\{\begin{aligned} 3 x-2 y &=8 \\\\-6 x+4 y &=16 \end{aligned}\right.$$

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

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. \(\left\\{\begin{array}{rr}{x^{2}+y=} & {8} \\ {x-2 y=} & {-6}\end{array}\right.\)

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l}{2 x+3 y>12} \\ {3 x-y<21}\end{array}\right.$$

\(15-22\) . Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$ \left[\begin{array}{rrr}{1} & {2} & {5} \\ {-2} & {-3} & {2} \\ {3} & {5} & {3}\end{array}\right] $$

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. $$ 5 A $$

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

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{aligned} 2 y+z &=4 \\ x+y \qquad=4 \\ 3 x+3 y-z &=10 \end{aligned}\right.$$

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

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

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. \(\left\\{\begin{array}{l}{x-y^{2}=-4} \\ {x-y=2}\end{array}\right.\)

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l}{y<\frac{1}{4} x+2} \\ {y \geq 2 x-5}\end{array}\right.$$

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

\(15-22\) . Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$ \left[\begin{array}{rrrr}{1} & {3} & {3} & {0} \\ {0} & {2} & {0} & {1} \\\ {-1} & {0} & {0} & {2} \\ {1} & {6} & {4} & {1}\end{array}\right] $$

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. $$ 3 B+2 C $$

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{aligned} x_{1}+2 x_{2}-x_{3} =9 \\ 2 x_{1} \qquad-x_{3}=-2 \\\ 3 x_{1}+5 x_{2}+2 x_{3} =22 \end{aligned}\right.$$

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. $$\left\\{\begin{aligned} x+4 y &=8 \\ 3 x+12 y &=2 \end{aligned}\right.$$

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

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. \(\left\\{\begin{aligned} x^{2}+y &=0 \\ x^{3}-2 x-y &=0 \end{aligned}\right.\)

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

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

\(15-22\) . Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$ \left[\begin{array}{rrrr}{1} & {2} & {0} & {2} \\ {3} & {-4} & {0} & {4} \\\ {0} & {1} & {6} & {0} \\ {1} & {0} & {2} & {0}\end{array}\right] $$

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. $$ C-5 A $$

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

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

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{array}{ll}{2 x_{1}+x_{2}} & {=7} \\ {2 x_{1}-x_{2}+x_{3}} & {=6} \\ {3 x_{1}-2 x_{2}+4 x_{3}} & {=11}\end{array}\right.$$

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

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. \(\left\\{\begin{aligned} x^{2}+y^{2} &=4 x \\ x &=y^{2} \end{aligned}\right.\)

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \\\ 3 x+2 y & \leq 9 \end{aligned}\right.$$

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

23-26 m Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{llll}{0} & {0} & {4} & {6} \\ {2} & {1} & {1} & {3} \\\ {2} & {1} & {2} & {3} \\ {3} & {0} & {1} & {7}\end{array}\right| $$

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. $$ 2 C-6 B $$

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{array}{l}{2 x-3 y-z=13} \\ {-x+2 y-5 z=6} \\ {5 x-y-z=49}\end{array}\right.$$

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. $$\left\\{\begin{aligned} 2 x-6 y &=10 \\\\-3 x+9 y &=-15 \end{aligned}\right.$$

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

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

19–40 Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x &>2 \\ y &<12 \\ 2 x-4 y &>8 \end{aligned}\right.$$

Find the partial fraction decomposition of the rational function. \(\frac{-3 x^{2}-3 x+27}{(x+2)\left(2 x^{2}+3 x-9\right)}\)

23-26 m Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{rrrr}{-2} & {3} & {-1} & {7} \\ {4} & {6} & {-2} & {3} \\\ {7} & {7} & {0} & {5} \\ {3} & {-12} & {4} & {0}\end{array}\right| $$

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. $$ D A $$

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. \(\left\\{\begin{array}{l}{3 x+4 y=10} \\ {7 x+9 y=20}\end{array}\right.\)

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. $$\left\\{\begin{aligned}2 x-3 y&=-8 \\ 14 x-21 &y=3 \end{aligned}\right.$$