In this module, we’ll learn how to solve problems where a quantity is being maximized or minimized. Additionally, this quantity is maximized or minimized in the presence of limitations.
As an example of this type of problem, let’s think about a craft brewery. Like most businesses, the brewery wants to produce different amounts of each beer so that its overall profit is as large as possible. You might think that they would produce as much as they can possible make and distribute. However, they are limited by the amount of ingredients they can buy and store as well as the amount of beer they can brew and ship. These limitations are called constraints. They prevent the brewery from ramping up production of each beer they produce. The solution to a problem like this, called a linear programming problem, is the production level of the different beers that maximizes the overall profit and meets all of the constraints.
Learning how to solves these types of problems will help you to master the objective for this chapter.
Solve linear programming problems by graphical and algebraic techniques.
Section 1 – Solving Systems of Linear Inequalities
Multiplicative inverses are useful for solving simple algebraic equations. For instance, the equation 2x = 6 may be solved by multiplying both sides of the equation by the multiplicative inverse of 2. When we do this, we get 1⁄2(2x) = 1⁄2(6). Since 1⁄2·2 = 1, the solution of the equation is x = 3.
In this section, we’ll learn how to write a system of linear equations as a matrix equation AX = B. This matrix equation is solved by multiplying both sides by an inverse matrix. This allows us to solve systems of equations with a strategy different from the matrix methods we used in Chapter 2.
For almost every real number, there is another number such that their product is equal to one. For instance, the product of 5 and 1⁄5 is 1. Numbers such as these are called multiplicative inverses. The exception to this rule is the number zero. It has no multiplicative inverse since any product with zero is zero.
Matrices share this property. In this section you’ll learn about the identity matrix, a matrix that plays the role that the number 1 plays for multiplicative inverses. In addition, you’ll learn how to find the inverse of a matrix.
When you add or subtract two matrices, you add or subtract the entries in two matrices of the same size. You might try to multiply two matrices by following a similar strategy. However, matrix multiplication is not carried out by multiplying the corresponding entries of two matrices of the same size.
Instead, matrix multiplication is carried out by multiplying the entries in the rows of a matrix by the entries in the columns of the other matrix. This might not seem to be a productive process. However, this process is very useful in many areas of business, economics, and science.
In this section you’ll learn how to carry out this process and apply it to several problems at Ed Magazine.
A matrix is a rectangular array of numbers. In Chapter 2 we used matrices to store the coefficients and constants in a system of linear equations. For instance, the system of linear equations
corresponds to the augmented matrix
In this section, we’ll expand on this usage to use matrices to solve business applications.
