Systems of Equations and Inequalities

58. Manufacturing Trucks Mike’s Toy Truck Company

manufactures two models of toy trucks, a standard model

and a deluxe model. Each standard model requires 2 hours

(hour) for painting and 3 hour for detail work; each deluxe

model requires 3 hour for painting and 4 hour for detail work.

Two painters and three detail workers are employed by the

company, and each works 40 hour per week.

(a) Using x to denote the number of standard-model trucks

and y to denote the number of deluxe-model trucks,

write a system of linear inequalities that describes the

possible number of each model of truck that can be

manufactured in a week.

(b) Graph the system and label the corner points.

Blending Coffee Bill’s Coffee House, a store that specializes in coffee, has available 75 pounds (lb) of A grade coffee and 120 lb of B grade coffee. These will be blended into 1-lb packages as follows: An economy blend that contains 4 ounces (oz) of A grade coffee and 12 oz of B grade coffee and a superior blend that contains 8 oz of A grade coffee and 8 oz of B grade coffee. 

(a) Using x to denote the number of packages of the economy blend and y to denote the number of packages of the superior blend, write a system of linear inequalities that describes the possible number of packages of each kind of blend.

(b) Graph the system and label the corner points. 

Mixed Nuts Nola’s Nuts, a store that specializes in selling nuts, has available 90 pounds (lb) of cashews and 120 lb of peanuts. These are to be mixed in 12-ounce (oz) packages as follows: a lower-priced package containing 8 oz of peanuts and 4 oz of cashews and a quality package containing 6 oz of peanuts and 6 oz of cashews. 

(a) Use x to denote the number of lower-priced packages and use y to denote the number of quality packages. Write a system of linear inequalities that describes the possible number of each kind of package. (b) Graph the system and label the corner points 

Transporting Goods A small truck can carry no more than 1600 pounds (lb) of cargo nor more than 150 cubic $\left({\mathrm{ft}}^3\right)$ of cargo. A printer weighs 20 lb and occupies 3 ${\mathrm{ft}}^3$ of space. A microwave oven weighs 30 lb and occupies 2 ${\mathrm{ft}}^3$ of space.

(a) Using x to represent the number of microwave ovens and y to represent the number of printers, write a system of linear inequalities that describes the number of ovens and printers that can be hauled by the truck. 

(b) Graph the system and label the corner points.

A linear programming problem requires that a linear expression, called the ______ _______, be maximized or minimized 

True or False. If a linear programming problem has a solution, it is located at a corner point of the graph of the feasible points.

Find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points. 

$z=x+y$

Find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points. 

$z=2x+3y$

Find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points. 

$z=x+10y$

Find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points. 

$z=10x+y$

Find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points. 

$z=5x+7y$

Find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points. 

$z=7x+5y$

Solve each linear programming problem.

Maximize $z=2x+y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 6\\ x+y\ge 1\end{array}\right.$

Solve each linear programming problem.

Maximize $z=x+3y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 3\\ x\le 5\\ y\le 7\end{array}\right.$

Solve each linear programming problem.

Minimize $2x+5y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ x\le 5\\ y\le 3\end{array}\right.$

Solve each linear programming problem.

Minimize $z=3x+4y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 2x+3y\ge 6\\ x+y\le 8\end{array}\right.$

Solve each linear programming problem.

Maximize $z=3x+5y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ 2x+3y\le 12\\ 3x+2y\le 12\end{array}\right.$

Solve each linear programming problem.

Maximize $z=5x+3y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ x+y\le 8\\ 2x+y\le 10\end{array}\right.$

Solve each linear programming problem.

Minimize $z=5x+4y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ 2x+3y\le 12\\ 3x+y\le 12\end{array}\right.$

Solve each linear programming problem.

Minimize $z=2x+3y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 3\\ x+y\le 9\\ x+3y\ge 6\end{array}\right.$

Solve each linear programming problem.

Minimize $5x+2y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 10\\ 2x+y\ge 10\\ x+2y\ge 10\end{array}\right.$

In problem 9-18,solve each linear programming problem.

Maximize $z=2x+4y$ subject to $x\ge 0,y\ge 0,2x+y\ge 4,x+y\le 9$

Farm Management A farmer has 70 acres of land available for planting either soybeans or wheat. The cost of preparing the soil, the workdays required, and the expected profit per acre planted for each type of crop are given in the following table:

The farmer cannot spend more than $\backslash (1800$ in preparation costs nor use more than a total of 120 workdays. How many acres of each crop should be planted to maximize the profit? What is the maximum profit? What is the maximum profit if the farmer is willing to spend no more than $\backslash )2400$ on preparation? 

Banquet Seating A banquet hall offers two types of tables for rent: 6-person rectangular tables at a cost of $\backslash (28$ each and 10 -person round tables at a cost of $\backslash )52$ each. Kathleen would like to rent the hall for a wedding banquet and needs tables for 250 people. The room can have a maximum of 35 tables, and the hall only has 15 rectangular tables available. How many of each type of table should be rented to minimize cost, and what is the minimum cost? 

Spring Break The student activities department of a community college plans to rent buses and vans for a spring-break trip. Each bus has 40 regular seats and 1 handicapped seat; each van has 8 regular seats and 3 handicapped seats. The rental cost is $\backslash (350$ for each van and $\backslash )975$ for each bus. If 320 regular and 36 handicapped seats are required for the trip, how many vehicles of each type should be rented to minimize cost? 

Return on Investment An investment broker is instructed by her client to invest up to $\backslash (20,000$, some in a junk bond yielding $9\%$ per annum and some in Treasury bills yielding $7\%$ per annum. The client wants to invest at least $\backslash )8000$ in T-bills and no more than $\$12000$ in the junk bond.
(a) How much should the broker recommend that the client place in each investment to maximize income if the client insists that the amount invested in T-bills must equal or exceed the amount placed in junk bonds?
(b) How much should the broker recommend that the client place in each investment to maximize income if the client insists that the amount invested in T-bills must not exceed the amount placed in junk bonds? 

Production Scheduling In a factory, machine 1 produces 8 -inch (in.) pliers at the rate of 60 units per hour (hr) and 6 -in. pliers at the rate of 70 units/hr. Machine 2 produces 8 -in. pliers at the rate of $40units/hr$ and 6-in. pliers at the rate of $20units/hr$. It costs $\backslash (50/hr$ to operate machine 1 , and machine 2 costs $\backslash )30/hr$ to operate. The production schedule requires that at least 240 units of 8-in. pliers and at least 140 units of 6-in. pliers be produced during each 10-hr day. Which combination of machines will cost the least money to operate?

Managing a Meat Market A meat market combines ground beef and ground pork in a single package for meat loaf. The ground beef is $75\%$ lean ( $75\%$ beef, $25\%$ fat ) and costs the market $\backslash (0.75$ per pound (lb). The ground pork is $60\%$ lean and costs the market $\backslash )0.45/lb$. The meat loaf must be at least $70\%$ lean. If the market wants to use at least $50/lb$ of its available pork, but no more than $200/lb$ of its available ground beef, how much ground beef should be mixed with ground pork so that the cost is minimized? 

Ice Cream The Mom and Pop Ice Cream Company makes    two kinds of chocolate ice cream: regular and premium. The properties of 1 gallon (gal) of each type are shown in the table:

In addition, current commitments require the company to make at least 1 gal of premium for every 4 gal of regular. Each day, the company has available 725 pounds (lb) of flavoring and $425lb$ of milk-fat products. If the company can ship no more than 3000 lb of product per day, how many gallons of each type should be produced daily to maximize profit? 

Financial Planning A retired couple has up to \($50000$ to place in fixed-income securities. Their financial adviser suggests two securities to them: one is an AAA bond that yields $8$% per annum; the other is a certificate of deposit (CD) that yields $4$%. After careful consideration of the alternatives, the couple decides to place at most \)$20000$ in the AAA bond and at least $$15000$ in the CD. They also instruct the financial adviser to place at least as much in the CD as in the AAA bond. How should the financial adviser proceed to maximize the return on their investment? 

Product Design An entrepreneur is having a design group produce at least six samples of a new kind of fastener that he wants to market. It costs \($9.00$ to produce each metal fastener and \)$4.00$ to produce each plastic fastener. He wants to have at least two of each version of the fastener and needs to have all the samples $24$ hours (hr) from now. It takes $4$ hr to produce each metal sample and $2$ hr to produce each plastic sample. To minimize the cost of the samples, how many of each kind should the entrepreneur order? What will be the cost of the samples? 

Animal Nutrition Kevin’s dog Amadeus likes two kinds of canned dog food. Gourmet Dog costs $40$ cents a can and has $20$ units of a vitamin complex; the calorie content is $75$ calories. Chow Hound costs $32$ cents a can and has $35$ units of vitamins and $50$ calories. Kevin likes Amadeus to have at least $1175$ units of vitamins a month and at least $2375$ calories during the same time period. Kevin has space to store only $60$ cans of dog food at a time. How much of each kind of dog food should Kevin buy each month to minimize his cost? 

Airline Revenue An airline has two classes of service: first class and coach. Management's experience has been that each aircraft should have at least $8$ but no more than $16$ first class seats and at least $80$ but not more than $120$ coach seats.

(a) If management decides that the ratio of first class to coach seats should never exceed $1:12$, with how many of each type of seat should an aircraft be configured to maximize revenue?

(b) If management decides that the ratio of first class to coach seats should never exceed $1:8$, with how many of each type of seat should an aircraft be configured to maximize revenue?

(c) If you were management, what would you do? 

[Hint: Assume that the airline charges \($C$ for a coach seat and \)$F$ for a first-class seat; $C>0,F>0$.] 

Explain in your own words what a linear programming problem is and how it can be solved.

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent. 

$\left\{\begin{array}{l}2x-y=5\\ 5x+2y=8\end{array}\right.$

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.  

$\left\{\begin{array}{l}3x-4y=4\\ x-3y=\frac{1}{2}\end{array}\right.$

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent. 

$\left\{\begin{array}{l}x-2y-4=0\\ 3x+2y-4=0\end{array}\right.$

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.  

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

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.   

$\left\{\begin{array}{l}x-3y+4=0\\ \frac{1}{2}x-\frac{3}{2}y+\frac{4}{3}=0\end{array}\right.$

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent. 

$\left\{\begin{array}{l}2x+3y-13=0\\ 3x-2y=0\end{array}\right.$

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.   

$\left\{\begin{array}{l}2x+5y=10\\ 4x+10y=20\end{array}\right.$

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.   

$\left\{\begin{array}{l}x+2y-z=6\\ 2x-y+3z=-13\\ 3x-2y+3z=-16\end{array}\right.$

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.   

$\left\{\begin{array}{l}2x-4y+z=-15\\ x+2y-4z=27\\ 5x-6y-2z=-3\end{array}\right.$

In Problems 1–10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.    

$\left\{\begin{array}{l}x-4y+3z=15\\ -3x+y-5z=-5\\ -7x-5y-9z=10\end{array}\right.$

Write the system of equations corresponding to the given augmented matrix.

$\left[\begin{array}{rrr}3& 2& 8\\ 1& 4& -1\end{array}\right]$

Write the system of equations corresponding to the given augmented matrix.

$\left[\begin{array}{rrrr}1& 2& 5& -2\\ 5& 0& -3& 8\\ 2& -1& 0& 0\end{array}\right]$

In Problems 13–16, use the following matrices to compute each expression.

$A=\left[\begin{array}{cc}1& 0\\ 2& 4\\ -1& 2\end{array}\right] B=\left[\begin{array}{ccc}4& -3& 0\\ 1& 1& -2\end{array}\right] C=\left[\begin{array}{cc}3& -4\\ 1& 5\\ 5& 2\end{array}\right]$

$A+C$

In Problems 13–16, use the following matrices to compute each expression.

$A=\left[\begin{array}{cc}1& 0\\ 2& 4\\ -1& 2\end{array}\right] B=\left[\begin{array}{ccc}4& -3& 0\\ 1& 1& -2\end{array}\right] C=\left[\begin{array}{cc}3& -4\\ 1& 5\\ 5& 2\end{array}\right]$

$6A$