Chapter 8

Solve each system by the substitution method. $$\left\\{\begin{array}{l}{x+y=2} \\\\{y=x^{2}-4}\end{array}\right.$$

Graph each inequality. $$x+2 y \leq 8$$

Determine if the given ordered triple is a solution of the system. $$\begin{aligned} &(2,-1,3)\\\&\left\\{\begin{aligned} x+y+z &=4 \\ x-2 y-z &=1 \\ 2 x-y-2 z &=-1 \end{aligned}\right. \end{aligned} $$

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{11 x-10}{(x-2)(x+1)}$$

Determine whether the given ordered pair is a solution of the system. \((2,3)\) \(\left\\{\begin{array}{l}{x+3 y=11} \\ {x-5 y=-13}\end{array}\right.\)

Solve each system by the substitution method. $$ \left\\{\begin{array}{l} {x-y=-1} \\ {y=x^{2}+1} \end{array}\right. $$

Graph each inequality. $$3 x-6 y \leq 12$$

Determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(5,-3,-2)\\\ &\left\\{\begin{aligned} x+y+z &=0 \\ x+2 y-3 z &=5 \\ 3 x+4 y+2 z &=-1 \end{aligned}\right. \end{aligned} $$

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x+7}{(x-1)(x+3)}$$

Determine whether the given ordered pair is a solution of the system. \((-3,5)\) \(\left\\{\begin{array}{l}{9 x+7 y=8} \\ {8 x-9 y=-69}\end{array}\right.\)

Solve each system by the substitution method. $$\left\\{\begin{array}{l}{x+y=2} \\\\{y=x^{2}-4 x+4}\end{array}\right.$$

Graph each inequality. $$x-2 y>10$$

Determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(4,1,2)\\\ &\left\\{\begin{aligned} x-2 y &=2 \\ 2 x+3 y &=11 \\ y-4 z &=-7 \end{aligned}\right. \end{aligned} $$

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{6 x^{2}-14 x-27}{(x+2)(x-3)^{2}}$$

Determine whether the given ordered pair is a solution of the system. \((2,5)\) \(\left\\{\begin{array}{r}{2 x+3 y=17} \\ {x+4 y=16}\end{array}\right.\)

Solve each system by the substitution method. $$\left\\{\begin{array}{l}{2 x+y=-5} \\\\{y=x^{2}+6 x+7}\end{array}\right.$$

Graph each inequality. $$2 x-y>4$$

Determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(-1,3,2)\\\ &\left\\{\begin{array}{r} {x-2 z=-5} \\ {y-3 z=-3} \\ {2 x-z=-4} \end{array}\right. \end{aligned} $$

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{3 x+16}{(x+1)(x-2)^{2}}$$

Determine whether the given ordered pair is a solution of the system. \((8,5)\) \(\left\\{\begin{array}{l}{5 x-4 y=20} \\ {3 y=2 x+1}\end{array}\right.\)

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned} &z=3 x+2 y\\\ &\left\\{\begin{array}{c} {x \geq 0, y \geq 0} \\ {2 x+y \leq 8} \\ {x+y \geq 4} \end{array}\right. \end{aligned}$$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}{x+y=4} \\ {y=3 x}\end{array}\right.\)

Graph each inequality. $$y \leq \frac{1}{3} x$$

Solve each system. $$ \left\\{\begin{array}{l} {x+y+2 z=11} \\ {x+y+3 z=14} \\ {x+2 y-z=5} \end{array}\right. $$

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x^{2}-6 x+7}{(x-1)\left(x^{2}+1\right)}$$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=2 x+3 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+y \leq 8} \\ {2 x+3 y \leq 12} \end{array}\right. \end{aligned} $$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}{x+y=6} \\ {y=2 x}\end{array}\right.\)

Graph each inequality. $$y \leq \frac{1}{4} x$$

Solve each system. $$ \left\\{\begin{array}{rrr} {2 x+y-2 z} & {=} & {-1} \\ {3 x-3 y-z} & {=} & {5} \\ {x-2 y+3 z} & {=} & {6} \end{array}\right. $$

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x^{2}-9 x+19}{(x-4)\left(x^{2}+5\right)}$$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=4 x+y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+3 y \leq 12} \\ {x+y \geq 3} \end{array}\right. \end{aligned} $$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}{x+3 y=8} \\ {y=2 x-9}\end{array}\right.\)

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

Solve each system. $$ \left\\{\begin{aligned} 4 x-y+2 z &=11 \\ x+2 y-z &=-1 \\ 2 x+2 y-3 z &=-1 \end{aligned}\right. $$

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{x^{3}+x^{2}}{\left(x^{2}+4\right)^{2}}$$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=x+6 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+y \leq 10} \\ {x-2 y \geq-10} \end{array}\right. \end{aligned} $$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}{2 x-3 y=-13} \\ {y=2 x+7}\end{array}\right.\)

Graph each inequality. $$y>3 x+2$$

$$ \left\\{\begin{aligned} x-y+3 z &=8 \\ 3 x+y-2 z &=-2 \\ 2 x+4 y+z &=0 \end{aligned}\right. $$

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{7 x^{2}-9 x+3}{\left(x^{2}+7\right)^{2}}$$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=3 x-2 y\\\ &\left\\{\begin{array}{l} {1 \leq x \leq 5} \\ {y \geq 2} \\ {x-y \geq-3} \end{array}\right. \end{aligned} $$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}{x=4 y-2} \\ {x=6 y+8}\end{array}\right.\)

Graph each inequality. $$x \leq 1$$

Solve each system. $$ \left\\{\begin{array}{l} {3 x+2 y-3 z=-2} \\ {2 x-5 y+2 z=-2} \\ {4 x-3 y+4 z=10} \end{array}\right. $$

write the partial fraction decomposition of each rational expression. $$\frac{x}{(x-3)(x-2)}$$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=5 x-2 y\\\ &\left\\{\begin{array}{l} {0 \leq x \leq 5} \\ {0 \leq y \leq 3} \\ {x+y \geq 2} \end{array}\right. \end{aligned} $$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}{x=3 y+7} \\ {x=2 y-1}\end{array}\right.\)

Graph each inequality. $$x \leq-3$$

Solve each system. $$ \left\\{\begin{array}{l} {2 x+3 y+7 z=13} \\ {3 x+2 y-5 z=-22} \\ {5 x+7 y-3 z=-28} \end{array}\right. $$

write the partial fraction decomposition of each rational expression. $$\frac{1}{x(x-1)}$$