Chapter 9

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

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

Graph the inequality. $$y<-x+5$$

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

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

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

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} 12 x+15 y &=-18 \\ 2 x+\frac{5}{2} y &=-3 \end{aligned}\right.$$

Express the rule in function notation. (For example, the rule "square, then subtract 5 " is expressed as the function \(f(x)=x^{2}-5\). Subtract \(5,\) then square

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

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

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

Graph the inequality. $$2 x-y \leq 8$$

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

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

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

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{aligned}x+y^{2} &=0 \\\2 x+5 y^{2} &=75\end{aligned}\right.$$

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

Express the rule in function notation. (For example, the rule "square, then subtract 5 " is expressed as the function \(f(x)=x^{2}-5\). Take the square root, add \(8,\) then multiply by \(\frac{1}{3}\)

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

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

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

Graph the inequality. $$3 x+4 y+12>0$$

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. $$\left[\begin{array}{lll} 2 & 1 & 2 \\ 6 & 3 & 4 \end{array}\right]\left[\begin{array}{rr} 1 & -2 \\ 3 & 6 \\ -2 & 0 \end{array}\right]$$

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

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

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. $$\left\\{\begin{array}{l} x-y=3 \\ x+3 y=7 \end{array}\right.$$

Express the function (or rule) in words. $$h(x)=x^{2}+2$$

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

Use back-substitution to solve the triangular system. $$\left\\{\begin{array}{r} 2 x-y+6 z=5 \\ y+4 z=0 \\ -2 z=1 \end{array}\right.$$

Graph the inequality. $$4 x+5 y<20$$

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

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

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

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

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. $$\left\\{\begin{array}{l} 2 x-3 y=9 \\ 4 x+3 y=9 \end{array}\right.$$

Express the function (or rule) in words. $$k(x)=\sqrt{x+2}$$

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

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

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

Graph the inequality. $$-x^{2}+y \geq 10$$

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. $$\left[\begin{array}{rr} 2 & -3 \\ 0 & 1 \\ 1 & 2 \end{array}\right]\left[\begin{array}{l} 5 \\ 1 \end{array}\right]$$

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

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

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. $$\left\\{\begin{array}{c} 3 x+2 y=0 \\ -x-2 y=8 \end{array}\right.$$

Evaluate the minor and cofactor using the matrix \(A\) $$A=\left[\begin{array}{rrr} 1 & 0 & \frac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{array}\right]$$ $$M_{12}, A_{12}$$

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

Use back-substitution to solve the triangular system. Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$\left\\{\begin{aligned} x-2 y-z &=4 & & \text { Eliminate the } x \text { -term } \\ x-y+3 z &=0 & & \text { from the second equation. } \\ 2 x+y+z &=0 & & \end{aligned}\right.$$

Graph the inequality. $$y>x^{2}+1$$