Why study math? Linear Systems and the Substitution Method – Part II

As we are quickly learning from my series of articles on lines and their applications, the power of these mathematical objects should not be taken for granted due to their simplicity. Lines, and more specifically linear systems, find important applications in the fields of telecommunications, signal processing and automatic control, the latter field dealing with such interesting things as ballistic missile programming, guidance and control. In the first article of this series, we examined how to solve a linear system by the substitution method. Here we will look at some basic problems that employ such linear systems.

Let’s look at the following example dealing with entrance fees to museums, say, the Natural History Museum in New York City. Suppose that in one day, this museum raised $1,590 from 321 people admitted to view its splendors. The price of each adult ticket is $6. Persons between the ages of 4-17 pay the $4 child admission. Let’s calculate how many of each type, adults and children, entered the museum.

Linear systems give us a neat way to solve this problem. To appreciate the power of linear systems and the substitution methodwhich we are going to use to solve this problem,

try to guess how you would solve this. You will quickly see that there is no convenient way to get the number of adults and the number of children who were admitted. However, by creating some linear models in system form, we can quickly arrive at the answer to this problem.

Let’s start with a verb model and then translate this to mathematics. This is a convenient and useful strategy that will allow us to solve the problem more easily. We have the information that the Number of Adults more him number of children is equal to 321. We also have that the number of adults multiplied by the price of an adult ticket more him number of children multiplied by the price of a child’s ticket is equal to the total amount raised.

If we let x represent the number of adults and y represent the number of children, we can translate the verbal model into a linear system of equations. Since the adult price is $6 and the child price is $4 and the total number of people attending is 321, we have

x + y = 321 and 6x + 4y = 1590. Notice that both equations are in standard form. We can easily take the first equation and put it into slope intercept written form

y = -x + 321 (moving x to the other side). We now substitute into the second equation to obtain

6x + 4(-x + 321) = 1590. Simplifying, we have 2x = 306 or x = 153. Using this value of x to get y, we have y = 168.

Thus, 153 adults and 168 children attended the museum. And because all the kids were so good at doing something educational with their parents, they all went home with great memories. Not a good way to spend the day!