Chapter 5

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

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

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 \(z=5 x-2 y\) \(\left\\{\begin{array}{l}0 \leq x \leq 5 \\ 0 \leq y \leq 3 \\ x+y \geq 2\end{array}\right.\)

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x y=-12 \\ x-2 y+14=0 \end{array}\right. $$

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x=3 y+7 \\ x=2 y-1 \end{array}\right. $$

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. $$ \frac{3 x+50}{(x-9)(x+2)} $$

Graph each inequality. $$y>1$$

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 \(z=4 x+2 y\) \(\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\ 2 x+3 y \leq 12 \\ 3 x+2 y \leq 12 \\ x+y \geq 2\end{array}\right.\)

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} y^{2}=x^{2}-9 \\ 2 y=x-3 \end{array}\right. $$

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{aligned} 5 x+2 y &=0 \\ x-3 y &=0 \end{aligned}\right. $$

Solve each system. $$\left\\{\begin{aligned} 2 x-4 y+3 z &=17 \\ x+2 y-z &=0 \\ 4 x-y-z &=6 \end{aligned}\right.$$

write the partial fraction decomposition of each rational expression. $$ \frac{5 x-1}{(x-2)(x+1)} $$

Graph each inequality. $$y>-3$$

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 \(z=2 x+4 y\) \(\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\ x+3 y \geq 6 \\ x+y \geq 3 \\\ x+y \leq 9\end{array}\right.\)

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x^{2}+y=4 \\ 2 x+y=1 \end{array}\right. $$

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 4 x+3 y=0 \\ 2 x-y=0 \end{array}\right. $$

Solve each system. $$\left\\{\begin{array}{r} x+\quad z=3 \\ x+2 y-z=1 \\ 2 x-y+z=3 \end{array}\right.$$

write the partial fraction decomposition of each rational expression. $$ \frac{7 x-4}{x^{2}-x-12} $$

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 \(z=10 x+12 y\) \(\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\ x+y \leq 7 \\ 2 x+y \leq 10 \\\ 2 x+3 y \leq 18\end{array}\right.\)

Graph each inequality. $$x^{2}+y^{2} \leq 1$$

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x y=3 \\ x^{2}+y^{2}=10 \end{array}\right. $$

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 2 x+5 y=-4 \\ 3 x-y=11 \end{array}\right. $$

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

write the partial fraction decomposition of each rational expression. $$ \frac{9 x+21}{x^{2}+2 x-15} $$

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 \(z=5 x+6 y\) \(\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\ 2 x+y \geq 10 \\ x+2 y \geq 10 \\ x+y \leq 10\end{array}\right.\)

Graph each inequality. $$x^{2}+y^{2} \leq 4$$

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x y=4 \\ x^{2}+y^{2}=8 \end{array}\right. $$

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 2 x+5 y=1 \\ -x+6 y=8 \end{array}\right. $$

Solve each system. $$\left\\{\begin{aligned} x+3 y+5 z &=20 \\ y-4 z &=-16 \\ 3 x-2 y+9 z &=36 \end{aligned}\right.$$

write the partial fraction decomposition of each rational expression. $$ \frac{4}{2 x^{2}-5 x-3} $$

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is 125 dollar for the rear-projection televisions and 200 dollar for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and let \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: - Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \(\cdot\) The cost to the manufacturer per unit is 600 for the rear-projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed 360,000 dollar Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be non negative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200),(450,100),\) and \((450,0)\).]

Graph each inequality. $$x^{2}+y^{2}>25$$

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x+y=1 \\ x^{2}+x y-y^{2}=-5 \end{array}\right. $$

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 2 x-3 y=8-2 x \\ 3 x+4 y=x+3 y+14 \end{array}\right. $$

Solve each system. $$\left\\{\begin{array}{rr} x+y & =-4 \\ y-z & =1 \\ 2 x+y+3 z & =-21 \end{array}\right.$$

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

a. A student earns 10 dollar per hour for tutoring and 7 dollar per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and let \(y=\) the number of hour search week spent as a teacher's aide. Write the objective function that models total weekly earnings. b. The student is bound by the following constraints: To have enough time for studies, the student can work no more than 20 hours per week. The tutoring center requires that each tutor spend at least three hours per week tutoring. The tutoring center requires that each tutor spend no more than eight hours per week tutoring. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17),\) and \((8,12)\).] Complete the missing portions of this statement: The student can earn the maximum amount per week by _____ hours per week and working as a teacher's aide for hours per week. The maximum amount that the student can earn each week is ______

Graph each inequality. $$x^{2}+y^{2}>36$$

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x+y=-3 \\ x^{2}+2 y^{2}=12 y+18 \end{array}\right. $$

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 3 x-4 y=x-y+4 \\ 2 x+6 y=5 y-4 \end{array}\right. $$

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

write the partial fraction decomposition of each rational expression. $$ \frac{4 x^{2}+13 x-9}{x(x-1)(x+3)} $$

Graph each inequality. $$(x-2)^{2}+(y+1)^{2}<9$$

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x+y=1 \\ (x-1)^{2}+(y+2)^{2}=10 \end{array}\right. $$

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} y=\frac{1}{3} x+\frac{2}{3} \\ y=\frac{5}{7} x-2 \end{array}\right. $$

Solve each system. $$\left\\{\begin{aligned} 3(2 x+y)+5 z &=-1 \\ 2(x-3 y+4 z) &=-9 \\ 4(1+x) &=-3(z-3 y) \end{aligned}\right.$$

write the partial fraction decomposition of each rational expression. $$ \frac{4 x^{2}-5 x-15}{x(x+1)(x-5)} $$

Graph each inequality. $$(x+2)^{2}+(y-1)^{2}<16$$

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 2 x+y=4 \\ (x+1)^{2}+(y-2)^{2}=4 \end{array}\right. $$