Chapter 10

1–14 Graph the inequality. $$x^{2}+y^{2} \leq 25$$

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

\(9-14\) Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{23}, A_{23} $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rr}{0.4} & {-1.2} \\ {0.3} & {0.6}\end{array}\right]\)

Solve the matrix equation for the unknown matrix \(X\) , or explain why no solution exists. $$\begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right]} & {D=\left[\begin{array}{rr}{10} & {20} \\\ {10} & {0}\end{array}\right]}\end{array}$$ $$ 2(B-X)=D $$

7–14 A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{rrrrr}{1} & {3} & {0} & {-1} & {0} \\ {0} & {0} & {1} & {2} & {0} \\ {0} & {0} & {0} & {0} & {1} \\ {0} & {0} & {0} & {0} & {0}\end{array}\right]$$

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. $$\left\\{\begin{array}{l}{-x+y=2} \\ 4 x-3 y=-3\end{array}\right.$$

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{array}{c}{3 x^{2}-y^{2}=11} \\ {x^{2}+4 y^{2}=8}\end{array}\right.\)

1–14 Graph the inequality. $$x^{2}+(y-1)^{2} \leq 1$$

Find the partial fraction decomposition of the rational function. \(\frac{x+6}{x(x+3)}\)

\(9-14\) Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{32}, A_{32} $$

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$ \left[\begin{array}{llllll} 1 & 3 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array}\right] $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{lll}{4} & {2} & {3} \\ {3} & {3} & {2} \\ {1} & {0} & {1}\end{array}\right]\)

7–14 A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{cccccc}{1} & {3} & {0} & {1} & {0} & {0} \\ {0} & {1} & {0} & {4} & {0} & {0} \\ {0} & {0} & {0} & {1} & {1} & {2} \\ {0} & {0} & {0} & {1} & {0} & {0}\end{array}\right]$$

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

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \left\\{\begin{aligned} x-4 y+z &=3 \\ y-3 z &=10 \\ 3 y-8 z &=24 \end{aligned}\right. $$ Eliminate the \(y\) -term from the third equation.

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{aligned} 2 x^{2}+4 y &=13 \\ x^{2}-y^{2} &=\frac{7}{2} \end{aligned}\right.\)

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

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

Solve the matrix equation for the unknown matrix \(X\) , or explain why no solution exists. $$\begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right]} & {D=\left[\begin{array}{rr}{10} & {20} \\\ {10} & {0}\end{array}\right]}\end{array}$$ $$ \frac{1}{5}(X+D)=C $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rrr}{2} & {4} & {1} \\ {-1} & {1} & {-1} \\ {1} & {4} & {0}\end{array}\right]\)

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. $$\left\\{\begin{aligned} x+2 y=7 \\ 5 x-y=2\end{aligned}\right.$$

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{aligned} x-2 y+z &=1 \\ y+2 z &=5 \\ x+y+3 z &=8 \end{aligned}\right.$$

Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+y+z &=4 \\ x+3 y+3 z &=10 \\ 2 x+y-z &=3 \end{aligned}\right. $$

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{aligned} x-y^{2}+3 &=0 \\ 2 x^{2}+y^{2}-4 &=0 \end{aligned}\right.\)

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

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

Solve the matrix equation for the unknown matrix \(X\) , or explain why no solution exists. $$\begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right]} & {D=\left[\begin{array}{rr}{10} & {20} \\\ {10} & {0}\end{array}\right]}\end{array}$$ $$ 2 A=B-3 X $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rrr}{5} & {7} & {4} \\ {3} & {-1} & {3} \\ {6} & {7} & {5}\end{array}\right]\)

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. $$\left\\{\begin{aligned}-4 x+12 y &=0 \\ 12 x+4 y &=160 \end{aligned}\right.$$

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

Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+y+z &=0 \\\\-x+2 y+5 z &=3 \\ 3 x-y &=6 \end{aligned}\right. $$

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{array}{c}{x^{2}-y^{2}=1} \\ {2 x^{2}-y^{2}=x+3}\end{array}\right.\)

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

15–18 An equation and its graph are given. Find an inequality whose solution is the shaded region. $$x^{2}+y^{2}=4$$

\(15-22\) . Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$ \left[\begin{array}{rrr}{1} & {3} & {7} \\ {2} & {0} & {-1} \\ {0} & {2} & {6}\end{array}\right] $$

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. $$ B+C $$a

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rrr}{1} & {2} & {3} \\ {4} & {5} & {-1} \\ {1} & {-1} & {-10}\end{array}\right]\)

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

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{aligned} x+y+z =2 \\ 2 x-3 y+2 z &=4 \\ 4 x+y-3 z &=1 \end{aligned}\right.$$

Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-& 4 z=1 \\ 2 x-y-6 z &=4 \\ 2 x+3 y-2 z &=8 \end{aligned}\right. $$

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. \(\left\\{\begin{array}{c}{2 x+y=-1} \\ {x-2 y=-8}\end{array}\right.\)

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

15–18 An equation and its graph are given. Find an inequality whose solution is the shaded region. $$y=x^{3}-4 x$$

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

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

Find the inverse of the matrix if it exists. \(\left[\begin{array}{lll}{2} & {1} & {0} \\ {1} & {1} & {4} \\ {2} & {1} & {2}\end{array}\right]\)

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. $$\left\\{\begin{aligned} 0.2 x-0.2 y &=-1.8 \\\\-0.3 x+0.5 y &=3.3 \end{aligned}\right.$$

15–24 The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. $$\left\\{\begin{aligned} x+y+z =4 \\\\-x+2 y+3 z =17 \\ 2 x-y \quad=-7 \end{aligned}\right.$$

Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-y+2 z &=2 \\ 3 x+y+5 z &=8 \\ 2 x-y-2 z &=-7 \end{aligned}\right. $$