Chapter 8

In Exercises \(5-20,\) graph each linear inequality. $$4 x-5 y \leq 20$$

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

Write the system of linear equations represented by the augmented matrix. Utilize the variables \(x, y,\) and \(z\). $$\left[\begin{array}{rrr|r} -1 & 2 & 4 & 4 \\ 7 & 9 & 3 & -3 \\ 4 & 6 & -5 & 8 \end{array}\right]$$

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} x-4 y=-7 \\ 3 x+8 y=19 \end{array}$$

Solve each system of linear equations. $$\begin{array}{rr} 2 x-3 y+z= & 1 \\ x+4 y-2 z= & 2 \\ 3 x-y+4 z= & -3 \end{array}$$

Solve each system of linear equations by substitution. $$\begin{aligned} &4 x-5 y=-7\\\ &3 x+8 y=30 \end{aligned}$$

In Exercises \(5-20,\) graph each linear inequality. $$4 x-2 y \geq 6$$

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

Write the system of linear equations represented by the augmented matrix. Utilize the variables \(x, y,\) and \(z\). $$\left[\begin{array}{rrr|r} -1 & 0 & 0 & 4 \\ 7 & 9 & 3 & -3 \\ 4 & 6 & -5 & 8 \end{array}\right]$$

Use Cramer's rule to solve each system of equations, if possible. $$\begin{aligned} &7 x-3 y=-29\\\ &5 x+2 y=0 \end{aligned}$$

Solve each system of linear equations. $$\begin{array}{rr} 3 x+2 y+z= & 4 \\ -4 x-3 y-z= & -15 \\ x-2 y+3 z= & 12 \end{array}$$

Solve each system of linear equations by substitution. $$\begin{array}{r} \frac{1}{3} x-\frac{1}{4} y=0 \\ -\frac{2}{3} x+\frac{3}{4} y=2 \end{array}$$

In Exercises \(5-20,\) graph each linear inequality. $$6 x-3 y \geq 9$$

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

Write the system of linear equations represented by the augmented matrix. Utilize the variables \(x, y,\) and \(z\). $$\left[\begin{array}{rrr|r} 2 & 3 & -4 & 6 \\ 7 & -1 & 5 & 9 \end{array}\right]$$

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rrr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1\end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2\end{array}\right]$$ $$A-B$$

Use Cramer's rule to solve each system of equations, if possible. $$\begin{aligned} &6 x-2 y=24\\\ &4 x+7 y=41 \end{aligned}$$

Solve each system of linear equations. $$\begin{array}{rr} 3 x-y+4 z= & 13 \\ -4 x-3 y-z= & -15 \\ x-2 y+3 z= & 12 \end{array}$$

Solve each system of linear equations by substitution. $$\begin{aligned} \frac{1}{5} x+\frac{2}{3} y &=10 \\ -\frac{1}{2} x-\frac{1}{6} y &=-7 \end{aligned}$$

In Exercises \(5-20,\) graph each linear inequality. $$6 x+4 y \leq 12$$

Find the partial-fraction decomposition for each rational function. $$\frac{9 x-11}{(x-3)(x+5)}$$

Write the system of linear equations represented by the augmented matrix. Utilize the variables \(x, y,\) and \(z\). $$\left[\begin{array}{ll|l} 1 & 0 & a \\ 0 & 1 & b \end{array}\right]$$

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} 3 x+5 y=16 \\ y-x=0 \end{array}$$

Solve each system of linear equations. $$\begin{array}{l} -x+2 y+z=-2 \\ 3 x-2 y+z=4 \\ 2 x-4 y-2 z=4 \end{array}$$

Solve each system of linear equations by substitution. $$\begin{aligned} &-3.9 x+4.2 y=15.3\\\ &-5.4 x+7.9 y=16.7 \end{aligned}$$

In Exercises \(5-20,\) graph each linear inequality. $$5 x-2 y \geq 10$$

Find the partial-fraction decomposition for each rational function. $$\frac{8 x-13}{(x-2)(x+1)}$$

Write the system of linear equations represented by the augmented matrix. Utilize the variables \(x, y,\) and \(z\). $$\left[\begin{array}{rrr|r} 3 & 0 & 5 & 1 \\ 0 & -4 & 7 & -3 \\ 2 & -1 & 0 & 8 \end{array}\right]$$

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{rr} -2 x-3 y= & 15 \\ 7 y+4 x= & -33 \end{array}$$

Solve each system of linear equations by substitution. $$\begin{aligned} &6.3 x-7.4 y=18.6\\\ &2.4 x+3.5 y=10.2 \end{aligned}$$

In Exercises \(21-50,\) graph each system of inequalities or indicate that the system has no solution. $$\begin{aligned} &y \geq x-1\\\ &y \leq x+1 \end{aligned}$$

Find the partial-fraction decomposition for each rational function. $$\frac{3 x+1}{(x-1)^{2}}$$

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. $$\left[\begin{array}{ll|l} 1 & 0 & 3 \\ 1 & 1 & 2 \end{array}\right]$$

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{r} 3 x-5 y=7 \\ -6 x+10 y=-21 \end{array}$$

Solve each system of linear equations. $$\begin{array}{rr} x-z-y= & 10 \\ 2 x-3 y+z= & -11 \\ y-x+z= & -10 \end{array}$$

Solve each system of linear equations by elimination. $$\begin{aligned} &x-y=-3\\\ &x+y=7 \end{aligned}$$

In Exercises \(21-50,\) graph each system of inequalities or indicate that the system has no solution. $$\begin{aligned} &y > x+1\\\ &y < x-1 \end{aligned}$$

Find the partial-fraction decomposition for each rational function. $$\frac{9 y-2}{(y-1)^{2}}$$

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. $$\left[\begin{array}{ll|l} 0 & 1 & 3 \\ 1 & 0 & 2 \end{array}\right]$$

Use Cramer's rule to solve each system of equations, if possible. $$\begin{aligned} &3 x-5 y=7\\\ &6 x-10 y=14 \end{aligned}$$

Solve each system of linear equations. $$\begin{array}{rr} 2 x+z+y= & -3 \\ 2 y-z+x= & 0 \\ x+y+2 z= & 5 \end{array}$$

Solve each system of linear equations by elimination. $$\begin{aligned} &x-y=-10\\\ &x+y=8 \end{aligned}$$

In Exercises \(21-50,\) graph each system of inequalities or indicate that the system has no solution. $$\begin{aligned} &y > 2 x+1\\\ &y < 2 x-1 \end{aligned}$$

Find the partial-fraction decomposition for each rational function. $$\frac{4 x-3}{x^{2}+6 x+9}$$

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. $$\left[\begin{array}{rrr|r} 1 & 0 & -1 & -3 \\ 0 & 1 & 3 & 14 \end{array}\right]$$

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rrr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1\end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2\end{array}\right]$$ $$24+3 B$$

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{rr} 2 x-3 y= & 4 \\ -10 x+15 y= & -20 \end{array}$$

Solve each system of linear equations. $$\begin{aligned} 3 x_{1}+x_{2}-x_{3} &=1 \\ x_{1}-x_{2}+x_{3} &=-3 \\ 2 x_{1}+x_{2}+x_{3} &=0 \end{aligned}$$

Solve each system of linear equations by elimination. $$\begin{aligned} &5 x+3 y=-3\\\ &3 x-3 y=-21 \end{aligned}$$

In Exercises \(21-50,\) graph each system of inequalities or indicate that the system has no solution. $$\begin{aligned} &y \leq 2 x-1\\\ &y \geq 2 x+1 \end{aligned}$$