Chapter 7

For each element in the second row of the given matrix, find its cofactor (See Example 3 .) $$\left[\begin{array}{rrr}-2 & 0 & 1 \\\1 & 2 & 0 \\\4 & 2 & 1\end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{4 x+2}{(x+2)(2 x-1)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 3 & 7 \\ 2 & 5 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{array}{r} x+5 y=6 \\ x=3 \end{array}$$

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ll}w & x \\ y & z\end{array}\right]=\left[\begin{array}{rr}3 & 2 \\ -1 & 4\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} &x+2 y+z=4\\\ &x+y=0\\\ &x-y=-2 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}&y=2 x+3\\\&3 x+4 y=78\end{aligned}$$

Graph each inequality. $$2 x-y>-1$$

For each element in the second row of the given matrix, find its cofactor (See Example 3 .) $$\left[\begin{array}{rrr}1 & -1 & 2 \\\1 & 0 & 2 \\\0 & -3 & 1\end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{x+2}{(x+1)(x-1)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} -5 & 3 \\ -8 & 5 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} 2 x+7 y &=1 \\ 5 x &=-15 \end{aligned}$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x-y-z &=6 \\ x+2 y &=1 \\ 2 x-y &=12 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}&y=4 x-6\\\&2 x+5 y=-8\end{aligned}$$

Graph each inequality. $$x+2 y \leq 6$$

For each element in the second row of the given matrix, find its cofactor (See Example 3 .) $$\left[\begin{array}{rrr}1 & 2 & -1 \\\2 & 3 & -2 \\\\-1 & 4 & 1\end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{x}{x^{2}+4 x-5}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} -1 & -2 \\ 3 & 4 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} 2 x+y+z &=3 \\ 3 x-4 y+2 z &=-7 \\ x+y+z &=2 \end{aligned}$$

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc}0 & 5 & x \\ -1 & 3 & y+2 \\ 4 & 1 & z\end{array}\right]=\left[\begin{array}{rrr}0 & w+3 & 6 \\ -1 & 3 & 0 \\ 4 & 1 & 8\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x-y+z &=1 \\ -x+y-z &=0 \\ -x-y+z &=-2 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}&3 x-2 y=12\\\&5 x=4-2 y\end{aligned}$$

Graph each inequality. $$x-y \geq 2$$

For each element in the second row of the given matrix, find its cofactor (See Example 3 .) $$\left[\begin{array}{rrr}2 & -1 & 4 \\\3 & 0 & 1 \\\\-2 & 1 & 4\end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{5 x-3}{(x+1)(x-3)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} -1 & 2 \\ -2 & -1 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} &4 x-2 y+3 z=4\\\ &3 x+5 y+z=7\\\ &5 x-y+4 z=7 \end{aligned}$$

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc}3+x & 4 & t \\ 5 & 8-w & y+1 \\ -4 & 3 & 2 r\end{array}\right]=\left[\begin{array}{ccc}9 & 4 & 6 \\ z+3 & w & 9 \\ p & q & r\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{c} x-y-z=6 \\ x+2 y+2 z=3 \\ 2 x+y+z=9 \end{array}$$

Solve each system by substitution. $$\begin{aligned}&8 x+3 y=2\\\&5 x=17+6 y\end{aligned}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}4 & -7 & 8 \\\2 & 1 & 3 \\\\-6 & 3 & 0\end{array}\right]$$

Graph each inequality. $$2 x+3 y \geq 4$$

Find the partial fraction decomposition for each rational expression. $$\frac{2 x}{(x+1)(x+2)^{2}}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} -6 & 4 \\ -3 & 2 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} x+y &=2 \\ 2 y+z &=-4 \\ z &=2 \end{aligned}$$

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc}z & 4 r & 8 s \\ 6 p & 2 & 5\end{array}\right]+\left[\begin{array}{ccc}-9 & 8 r & 3 \\ 2 & 5 & 4\end{array}\right]=\left[\begin{array}{ccc}2 & 36 & 27 \\ 20 & 7 & 12 a\end{array}\right]$$

Solve each system by substitution. $$\begin{aligned}&4 x-5 y=-11\\\&2 x+y=5\end{aligned}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}8 & -2 & -4 \\\7 & 0 & 3 \\\5 & -1 & 2\end{array}\right]$$

Graph each inequality. $$4 y-3 x<5$$

Find the partial fraction decomposition for each rational expression. $$\frac{2}{x^{2}(x+3)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} 5 & 10 \\ -3 & -6 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} x &=6 \\ y+2 z &=2 \\ x-3 z &=6 \end{aligned}$$

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc}a+2 & 1 & 5 m \\ 8 k & 0 & 3\end{array}\right]+\left[\begin{array}{ccc}3 a & 2 z & 5 m \\ 2 k & 5 & 6\end{array}\right]=\left[\begin{array}{ccc}10 & -14 & 80 \\ 10 & 5 & 9\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{l} 2 x+y+z=9 \\ -x-y+z=1 \\ 3 x-y+z=9 \end{array}$$

Solve each system by substitution. $$\begin{aligned}&7 x-y=-10\\\&3 y-x=10\end{aligned}$$

Find the partial fraction decomposition for each rational expression. $$\frac{4}{x(1-x)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 0.6 & 0.2 \\ 0.5 & 0.1 \end{array}\right]$$

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{rr|r} 2 & 1 & 1 \\ 3 & -2 & -9 \end{array}\right]$$

Your friend missed the lecture on adding matrices. In your own words, explain to her how to add two matrices.

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{r} x+3 y+4 z=14 \\ 2 x-3 y+2 z=10 \\ 3 x-y+z=9 \end{array}$$