experiment

Linear Programming – Real-World Systems of Linear Inequalities
Entire careers are devoted to using systems of inequalities to ensure a company is making the most profit by producing the right combination of items or is spending the least amount of money to make certain items. **Linear programming** is the mathematical process of analyzing a system of inequalities to make the best decisions given the constraints of the situation. The goal is to locate the feasible region of the system and use it to answer a profitability, or **optimization**, question. This theorem provides an important piece of information. While the individual colors of the inequalities will overlap, providing an infinite number of possible combinations, only the vertices will provide the maximum (or minimum) solutions to the optimization equation. Let’s go back to the situation presented in the chapter opener. James is trying to expand his pastry business to include cupcakes and personal cakes. He has 40 hours available to decorate the new items and can use no more than 22 pounds of cake mix. Each personal cake requires 2 pounds of cake mix and 2 hours to decorate. Each cupcake order requires one pound of cake mix and 4 hours to decorate. If he can sell each personal cake for $14.99 and each cupcake order for $16.99, how many personal cakes and cupcake orders should James make to make the most revenue? There are four inequalities in this situation. First, state the variables. Let //the number of personal cakes// and  //the number of cupcake orders//. Translate this into a system of inequalities. – This is the amount of available cake mix. – This is the available time to decorate. – You cannot make negative personal cakes. – You cannot make negative cupcake orders. Now graph each inequality and determine the feasible region. The feasible region has four vertices {(0, 0),(0, 10),(11, 0),(8, 6)}. According to our theorem, the optimization answer will **only occur** at one of these vertices. Write the optimization equation: How much of each type of order should James make to bring in the most revenue? Substitute each ordered pair to determine which makes the most money [[image:http://www.ck12.org/ck12/ucs/?blockmath=%280%2C0%29%20%26%20%5Crightarrow%20%5C%240.00%5C%5C%0A%280%2C10%29%20%26%20%5Crightarrow%2014.99%280%29%2B16.99%2810%29%3D%5C%24169.90%5C%5C%0A%2811%2C0%29%20%26%5Crightarrow%2014.99%2811%29%2B16.99%280%29%3D%5C%24164.89%5C%5C%0A%288%2C6%29%20%26%20%5Crightarrow%2014.99%288%29%2B16.99%286%29%3D%5C%24221.86 caption="(0,0) & \rightarrow \$0.00\\ (0,10) & \rightarrow 14.99(0)+16.99(10)=\$169.90\\ (11,0) &\rightarrow 14.99(11)+16.99(0)=\$164.89\\ (8,6) & \rightarrow 14.99(8)+16.99(6)=\$221.86"]] To make the most revenue, James should make 8 personal cakes and 6 cupcake orders. For more help with applying systems of linear inequalities, watch this video by [] - PH School.
 * Constraints** are the particular restrictions of a situation due to time, money, or materials.
 * Theorem:** The maximum or minimum values of an optimization equation occur at the vertices of the feasible region – at the points where the boundary lines intersect.

Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. [| CK-12 Basic Algebra: Systems of Linear Inequalities] (8:52)

Find the solution region of the following systems of inequalities. > -5x\ge16-8y"]] > 2y-x \ge -10"]] > x+4y \le 6\!\\ > 3x+y \ge -4"]] > y\ge\frac{1}{4} x-3\\ > y<\frac{13}{8} x+8 \end{cases}"]] > y\le\frac{3}{4} x-5\\ > y\ge -2x+2 \end{cases}"]] > y>-x+1\\ > y>\frac{1}{4} x+6 \end{cases}"]] > y>-\frac{1}{2} x+4\\ > x<-4 \end{cases}"]] > y\le6\\ > y>\frac{1}{4} x+6 \end{cases}"]] Write the system of inequalities for each feasible region pictured below. Given the following constraints find the maximum and minimum values for: > 12x+8y=24\\ > -6x+3y=9 \end{cases}"]].
 * 1) What is //linear programming//?
 * 2) What is the //feasible region// of a system of inequalities?
 * 3) How do constraints affect the feasible region?
 * 4) What is an optimization equation? What is its purpose?
 * 5) You have graphed a feasible region. Where are the maximum (or minimum) points of the optimization equation located?
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=4y-5x%3C8%5C%21%5C%5C%0A-5x%5Cge16-8y caption="4y-5x<8\!\\
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=5x-y%20%5Cge5%5C%21%5C%5C%0A2y-x%20%5Cge%20-10 caption="5x-y \ge5\!\\
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=2x-3y%20%5Cle%2021%5C%21%5C%5C%0Ax%2B4y%20%5Cle%206%5C%21%5C%5C%0A3x%2By%20%5Cge%20-4 caption="2x-3y \le 21\!\\
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=%5Cbegin%7Bcases%7D%0Ay%5Cge%5Cfrac%7B1%7D%7B4%7D%20x-3%5C%5C%0Ay%3C%5Cfrac%7B13%7D%7B8%7D%20x%2B8%20%5Cend%7Bcases%7D caption="\begin{cases}
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=%5Cbegin%7Bcases%7D%0Ay%5Cle%5Cfrac%7B3%7D%7B4%7D%20x-5%5C%5C%0Ay%5Cge%20-2x%2B2%20%5Cend%7Bcases%7D caption="\begin{cases}
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=%5Cbegin%7Bcases%7D%0Ay%3E-x%2B1%5C%5C%0Ay%3E%5Cfrac%7B1%7D%7B4%7D%20x%2B6%20%5Cend%7Bcases%7D caption="\begin{cases}
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=%5Cbegin%7Bcases%7D%0Ay%3E-%5Cfrac%7B1%7D%7B2%7D%20x%2B4%5C%5C%0Ax%3C-4%20%5Cend%7Bcases%7D caption="\begin{cases}
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=%5Cbegin%7Bcases%7D%0Ay%5Cle6%5C%5C%0Ay%3E%5Cfrac%7B1%7D%7B4%7D%20x%2B6%20%5Cend%7Bcases%7D caption="\begin{cases}
 * 1) [[image:http://www.ck12.org/ck12/images?id=116134 width="175px"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=116151 width="175px"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=116114 width="175px"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=z%3D-x%2B5y%5C%21%5C%5Cx%2B3y%20%5Cle%200%5C%21%5C%5Cx-y%20%5Cge%200%5C%21%5C%5C3x-7y%20%5Cle%2016 caption="z=-x+5y\!\\x+3y \le 0\!\\x-y \ge 0\!\\3x-7y \le 16"]]
 * 2) //Find the maximum and minimum value of// [[image:http://www.ck12.org/ck12/ucs/?math=z%3D2x%2B5y caption="z=2x+5y"]] //given the constraints//.[[image:http://www.ck12.org/ck12/ucs/?blockmath=2x-y%20%26%20%5Cle%2012%5C%5C4x%2B3y%20%26%20%5Cge%200%5C%5Cx-y%20%26%20%5Cle%206 caption="2x-y & \le 12\\4x+3y & \ge 0\\x-y & \le 6"]]
 * 3) In Andrew’s Furniture Shop, he assembles both bookcases and TV cabinets. Each type of furniture takes him about the same time to assemble. He figures he has time to make at most 18 pieces of furniture by this Saturday. The materials for each bookcase cost him $20.00 and the materials for each TV stand cost him $45.00. He has $600.00 to spend on materials. Andrew makes a profit of $60.00 on each bookcase and a profit of $100.00 for each TV stand. Find how many of each piece of furniture Andrew should make so that he maximizes his profit.
 * 4) You have $10,000 to invest, and three different funds from which to choose. The municipal bond fund has a 5% return, the local bank's CDs have a 7% return, and a high-risk account has an expected 10% return. To minimize risk, you decide not to invest any more than $1,000 in the high-risk account. For tax reasons, you need to invest at least three times as much in the municipal bonds as in the bank CDs. Assuming the year-end yields are as expected, what are the optimal investment amounts?
 * Mixed Review**
 * 1) Solve by elimination [[image:http://www.ck12.org/ck12/ucs/?math=%5Cbegin%7Bcases%7D%0A12x%2B8y%3D24%5C%5C%0A-6x%2B3y%3D9%20%5Cend%7Bcases%7D caption="\begin{cases}
 * 1) Solve [[image:http://www.ck12.org/ck12/ucs/?math=36%3D%7C5t-6%7C caption="36=|5t-6|"]].
 * 2) Determine the intercepts of [[image:http://www.ck12.org/ck12/ucs/?math=y%3D-%5Cfrac%7B5%7D%7B6%7D%20x-3 caption="y=-\frac{5}{6} x-3"]].
 * 3) Jerry’s aunt repairs upholstery. For three hours' worth of work, she charges $145. For nine hours of work, she charges $355. Assuming this relationship is linear, write the equation for this line in point-slope form. How much would Jerry’s aunt charge for 1.25 hours worth of work?
 * 4) Translate into an algebraic sentence: “Yoder is four years younger than Kate. Kate is six years younger than Dylan. Dylan is 20.” How old is each person?