Chapter 3

Teams chosen from 30 forest rangers and 16 trainees are planting trees. An experienced team consisting of two rangers can plant 500 trees per week. A training team consisting of one ranger and two trainees can plant 200 trees per week. $$\begin{array}{|c|c|c|c|}\hline \text { Number of Teams } & {x} & {y} & {x+y} \\\ \hline \text { Number of Rangers } & {2 x} & {y} & {30} \\ \hline \text { Number of Trainees } & {0} & {2 y} & {16} \\ \hline \text { Number of Trees Planted } & {500 x} & {200 y} & {500 x+200 y} \\ \hline\end{array}$$ a. Write an objective function and constraints for a linear program that models the problem. b. How many of each type of team should be formed to maximize the number of trees planted? How many trainees are used in this solution? How many trees are planted? c. Find a solution that uses all the trainees. How many trees will be planted in this case?

Solve each system by substitution. Check your answers. \(\left\\{\begin{array}{l}{-6=3 x-6 y} \\ {4 x=4+5 y}\end{array}\right.\)

Solve each system of inequalities by graphing. $$ \left\\{\begin{aligned} x+2 y & \leq 10 \\ x+y & \leq 3 \end{aligned}\right. $$

Graph each point in coordinate space. $$ (1,1,0) $$

Solve each system by substitution. Check your answers. $$ \left\\{\begin{array}{l}{5 r-4 s-3 t=3} \\ {t=s+r} \\ {r=3 s+1}\end{array}\right. $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{-x-y \leq 2} \\ {y-2 x>1}\end{array}\right. $$

Graph each point in coordinate space. $$ (0,-2,2) $$

Solve each system by substitution. Check your answers. $$ \left\\{\begin{array}{l}{13=3 x-y} \\ {4 y-3 x+2 z=-3} \\ {z=2 x-4 y}\end{array}\right. $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y>-2 x} \\ {2 x-y \geq 2}\end{array}\right. $$

Solve each system by substitution. Check your answers. $$ \left\\{\begin{aligned} x+3 y-z &=-4 \\ 2 x-y+2 z &=13 \\ 3 x-2 y-z &=-9 \end{aligned}\right. $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{c \geq d-3} \\ {c<\frac{1}{2} d+3}\end{array}\right. $$

Graph each system of constraints. Name all vertices. Then find the values of \(x\) and \(y\) that maximize or minimize the objective function. Find the maximum or minimum value. \(\left\\{\begin{aligned} 3 x+y & \leq 7 \\ x+2 y & \leq 9 \\ x \geq 0, y & \geq 0 \end{aligned}\right.\) Maximum for \(P=2 x+y\)

A youth group with 26 members is going skiing. Each of the five chaperones will drive a van or a sedan. The vans can seat seven people, and the sedans can seat five people. How many of each type of vehicle could transport all 31 people to the ski area in one trip?

Without graphing, classify each system as independent, dependent, or inconsistent. $$ \left\\{\begin{aligned}-3 x+y &=4 \\ x-\frac{1}{3} y &=1 \end{aligned}\right. $$

Solve each system by substitution. Check your answers. $$ \left\\{\begin{aligned} x-4 y+z &=6 \\ 2 x+5 y-z &=7 \\ 2 x-y-z &=1 \end{aligned}\right. $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{2 x+y<1} \\ {-y+3 x<1}\end{array}\right. $$

Suppose you have a part-time job delivering packages. Your employer pays you at a flat rate of \(\$ 7\) per hour. You discover that a competitor pays employees \(\$ 2\) per hour plus \(\$ .35\) per delivery. a. Write a system of equations to model the pay \(p\) for \(d\) deliveries. Assume a four-hour shift. b. How many deliveries would the competitor's employees have to make in four hours to earn the same pay you earn in a four-hour shift?

Without graphing, classify each system as independent, dependent, or inconsistent. $$ \left\\{\begin{aligned} 4 x+8 y &=12 \\ x+2 y &=-3 \end{aligned}\right. $$

Solve each system by substitution. Check your answers. $$ \left\\{\begin{aligned} x-y+2 z &=7 \\ 2 x+y+z &=8 \\ x &-z=5 \end{aligned}\right. $$

Fund-Raising You want to bake at least 6 and at most 11 loaves of bread for a bake sale. You want at least twice as many loaves of banana bread as nut bread. Write and graph a system of inequalities to model the situation.

Graph each system of constraints. Name all vertices. Then find the values of \(x\) and \(y\) that maximize or minimize the objective function. Find the maximum or minimum value. \(\left\\{\begin{array}{l}{x+y \leq 11} \\ {2 y \geq x} \\ {x \geq 0, y \geq 0}\end{array}\right.\) Maximum for \(P=3 x+2 y\)

A boat can travel 24 \(\mathrm{mi}\) in 3 \(\mathrm{h}\) when traveling with a current. Against the same current, it can travel only 16 \(\mathrm{mi}\) in 4 \(\mathrm{h}\) . Find the rate of the current and the rate of the boat in still water.

Without graphing, classify each system as independent, dependent, or inconsistent. $$ \left\\{\begin{array}{l}{y=2 x-1} \\ {y=-2 x+5}\end{array}\right. $$

Solve each system by substitution. Check your answers. $$ \left\\{\begin{array}{l}{x+y+z=2} \\ {x+2 z=5} \\ {2 x+y-z=-1}\end{array}\right. $$

Graph each system of constraints. Name all vertices. Then find the values of \(x\) and \(y\) that maximize or minimize the objective function. Find the maximum or minimum value. \(\left\\{\begin{array}{c}{2 x+y \leq 300} \\ {x+y \leq 200} \\ {x \geq 0, y \geq 0}\end{array}\right.\) Maximum for \(P=x+2 y\)

The measure of one acute angle of a right triangle is \(30^{\circ}\) more than twice the measure of the other acute angle. Find the measures of the angles.

Solve each system by substitution. Check your answers. $$ \left\\{\begin{array}{r}{5 x-y+z=4} \\ {x+2 y-z=5} \\ {2 x+3 y-3 z=5}\end{array}\right. $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y>4} \\ {y<|x-1|}\end{array}\right. $$

Solve each system by elimination. \(\left\\{\begin{array}{l}{x+y=12} \\ {x-y=2}\end{array}\right.\)

Without graphing, classify each system as independent, dependent, or inconsistent. $$ \left\\{\begin{array}{l}{2 y=5 x+6} \\ {-10 x+4 y=8}\end{array}\right. $$

Finance A company placed \(\$ 1,000,000\) in three different accounts. It placed part in short-term notes paying 4.5\(\%\) per year, twice as much in government bonds paying \(5 \%,\) and the rest in utility bonds paying 4\(\%\) . The income after one year was \(\$ 45,500 .\) How much did the company place in each account?

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y<-\frac{1}{3} x+1} \\ {y>|2 x-1|}\end{array}\right. $$

Solve each system by elimination. \(\left\\{\begin{array}{l}{x+2 y=10} \\ {x+y=6}\end{array}\right.\)

Without graphing, classify each system as independent, dependent, or inconsistent. $$ \left\\{\begin{aligned} x-3 y &=2 \\ 4 x-12 y &=8 \end{aligned}\right. $$

Finance A company placed \(\$ 1,000,000\) in three different accounts. It placed part in short-term notes paying 4.5\(\%\) per year, twice as much in government bonds paying \(5 \%,\) and the rest in utility bonds paying 4\(\%\) . The income after one year was \(\$ 45,500 .\) How much did the company place in each account?

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y>x-2} \\ {y \geq|x+2|}\end{array}\right. $$

Solve each system by elimination. \(\left\\{\begin{aligned} 3 a+4 b &=9 \\\\-3 a-2 b &=-3 \end{aligned}\right.\)

Without graphing, classify each system as independent, dependent, or inconsistent. $$ \left\\{\begin{array}{l}{x+4 y=12} \\ {2 x-8 y=4}\end{array}\right. $$

A change machine contains nickels, dimes, and quarters. There are 75 coins in the machine, and the value of the coins is \(\$ 7.25\) . There are 5 times as many nickels as dimes. Find the number of coins of each type in the machine.

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y \leq-\frac{4}{3} x} \\ {y \geq-|x|}\end{array}\right. $$

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

Without graphing, classify each system as independent, dependent, or inconsistent. $$ \left\\{\begin{array}{l}{4 x+8 y=-6} \\ {6 x+12 y=-9}\end{array}\right. $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{3 y<-x-1} \\ {y \leq|x+1|}\end{array}\right. $$

Solve each system by elimination. \(\left\\{\begin{array}{l}{2 w+5 y=-24} \\ {3 w-5 y=14}\end{array}\right.\)

Without graphing, classify each system as independent, dependent, or inconsistent. $$ \left\\{\begin{array}{l}{4 y-2 x=6} \\ {8 y=4 x-12}\end{array}\right. $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y>-2} \\ {y \leq-|x-3|}\end{array}\right. $$

Solve each system by elimination. \(\left\\{\begin{aligned} 3 u+3 v &=15 \\\\-2 u+3 v &=-5 \end{aligned}\right.\)

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{-2 x+y>3} \\ {y \leq-|x+4|}\end{array}\right. $$

Which point maximizes \(N=4 x+3 y\) and lies within the feasible region of the constraints at the right? \(\left\\{\begin{array}{l}{y \leq 9} \\ {2 x+2 y \leq 18} \\ {x \leq 3}\end{array}\right.\) A. \((0,0)\) B. \((9,0)\) C. \((3,6)\) D. \((0,9)\)

Solve each system by elimination. \(\left\\{\begin{array}{l}{x+3 y=11} \\ {x+4 y=14}\end{array}\right.\)