Chapter 11

\(7-12=\) Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} x-2 y+4 z &=3 \\ y+2 z &=7 \\ z &=2 \end{aligned}\right. $$

\(5-8\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{array}{c}{x-y=2} \\ {2 x+3 y=9}\end{array}\right. $$

Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{-2} & {1} \\ {3} & {-2}\end{array}\right] $$

\(3-8=\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{array}{c}{x^{2}-y=1} \\ {2 x^{2}+3 y=17}\end{array}\right. $$

State the dimension of the matrix. $$ \left[\begin{array}{r}{-3} \\ {0} \\ {1}\end{array}\right] $$

\(3-12\) . Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$ \frac{1}{x^{4}-1} $$

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{lll}{0} & {1} & {1} \\ {1} & {1} & {0}\end{array}\right]-\left[\begin{array}{lll}{2} & {1} & {-1} \\ {1} & {3} & {-2}\end{array}\right] $$

\(7-12=\) Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} x+y-3 z &=8 \\ y-3 z &=5 \\ z &=-1 \end{aligned}\right. $$

\(5-8\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{2 x+y=7} \\ {x+2 y=2}\end{array}\right. $$

\(9-14\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{x^{2}-2 y=1} \\ {x^{2}+5 y=29}\end{array}\right. $$

State the dimension of the matrix. $$ \left[\begin{array}{lll}{1} & {4} & {7}\end{array}\right] $$

\(3-12\) . Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$ \frac{x^{3}-4 x^{2}+2}{\left(x^{2}+1\right)\left(x^{2}+2\right)} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rr}{-3} & {-5} \\ {2} & {3}\end{array}\right] $$

Perform the matrix operation, or if it is impossible, explain why. $$ 3\left[\begin{array}{rr}{1} & {2} \\ {4} & {-1} \\ {1} & {0}\end{array}\right] $$

\(7-12=\) Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} x+2 y+z &=7 \\\\-y+3 z &=9 \\ 2 z &=6 \end{aligned}\right. $$

\(9-12\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{aligned} 3 x+4 y &=10 \\ x-4 y &=-2 \end{aligned}\right. $$

\(9-14\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{3 x^{2}+4 y=17} \\ {2 x^{2}+5 y=2}\end{array}\right. $$

State the dimension of the matrix. $$ \left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right] $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{ll}{3} & {4} \\ {7} & {9}\end{array}\right] $$

Perform the matrix operation, or if it is impossible, explain why. $$ 2\left[\begin{array}{lll}{1} & {1} & {0} \\ {1} & {0} & {1} \\ {0} & {1} & {1}\end{array}\right]+\left[\begin{array}{ll}{1} & {1} \\ {2} & {1} \\ {3} & {1}\end{array}\right] $$

\(9-12\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{2 x+5 y=15} \\ {4 x+y=21}\end{array}\right. $$

Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{cc}{\frac{1}{2}} & {\frac{1}{8}} \\ {1} & {\frac{1}{2}}\end{array}\right] $$

\(3-16=\) Graph the inequality. $$ 4 x+5 y<20 $$

\(9-14\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{c}{3 x^{2}-y^{2}=11} \\ {x^{2}+4 y^{2}=8}\end{array}\right. $$

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. $$ \left[\begin{array}{rrr}{1} & {0} & {-3} \\ {0} & {1} & {5}\end{array}\right] $$

\(3-12\) . Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$ \frac{x^{3}+x+1}{x(2 x-5)^{3}\left(x^{2}+2 x+5\right)^{2}} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rr}{2} & {5} \\ {-5} & {-13}\end{array}\right] $$

\(7-12=\) Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} 2 x-y+6 z &=5 \\ y+4 z &=0 \\\\-2 z &=1 \end{aligned}\right. $$

\(9-12\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{c}{x+2 y=5} \\ {2 x+3 y=8}\end{array}\right. $$

Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{2.2} & {-1.4} \\ {0.5} & {1.0}\end{array}\right] $$

\(9-14\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{c}{2 x^{2}+4 y=13} \\\ {x^{2}-y^{2}=\frac{7}{2}}\end{array}\right. $$

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. $$ \left[\begin{array}{rrr}{1} & {3} & {-3} \\ {0} & {1} & {5}\end{array}\right] $$

\(3-12\) . Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$ \frac{1}{\left(x^{3}-1\right)\left(x^{2}-1\right)} $$

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rr}{-7} & {4} \\ {8} & {-5}\end{array}\right] $$

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rrr}{2} & {1} & {2} \\ {6} & {3} & {4}\end{array}\right]\left[\begin{array}{rr}{1} & {-2} \\ {3} & {6} \\ {-2} & {0}\end{array}\right] $$

\(7-12=\) Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} 4 x+3 z &=10 \\ 2 y-z &=-6 \\ \frac{1}{2} z &=4 \end{aligned}\right. $$

\(9-12\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{4 x-3 y=11} \\ {8 x+4 y=12}\end{array}\right. $$

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{11}, A_{11} $$

\(3-16=\) Graph the inequality. $$ y>x^{2}+1 $$

\(9-14\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{aligned} x-y^{2}+3 &=0 \\ 2 x^{2}+y^{2}-4 &=0 \end{aligned}\right. $$

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. $$ \left[\begin{array}{llll}{1} & {2} & {8} & {0} \\ {0} & {1} & {3} & {2} \\\ {0} & {0} & {0} & {0}\end{array}\right] $$

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

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rr}{6} & {-3} \\ {-8} & {4}\end{array}\right] $$

\(13-16\) . Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \begin{array}{l}{\left\\{\begin{array}{l}{x-2 y-z=4} \\ {x-y+3 z=0} \\ {2 x+y+z=0}\end{array}\right.} \\ {\text { Eliminate the } x \text { -term }} \\\ {\text { rom the second equation. }}\end{array} $$

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{1} & {2} \\ {-1} & {4}\end{array}\right]\left[\begin{array}{rrr}{1} & {-2} & {3} \\ {2} & {2} & {-1}\end{array}\right] $$

\(13-14\) . Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$ \left\\{\begin{aligned} 2 x+y &=-1 \\ x-2 y &=-8 \end{aligned}\right. $$

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{33}, A_{33} $$

\(3-16=\) Graph the inequality. $$ x^{2}+y^{2} \geq 9 $$

\(9-14\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{c}{x^{2}-y^{2}=1} \\ {2 x^{2}-y^{2}=x+3}\end{array}\right. $$

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. $$ \left[\begin{array}{rrrr}{1} & {0} & {-7} & {0} \\ {0} & {1} & {3} & {0} \\\ {0} & {0} & {0} & {1}\end{array}\right] $$