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 because of their simplicity. Lines and more specifically, linear systems, find important applications in the fields of telecommunications, signal processing, and automatic control, the last field of which deals with such interesting things as the programming, guidance, and control of ballistic missiles. In the first article in this series, we examined how to solve a linear system by the method of substitution. Here we will look at some basic problems which employ such linear systems.

Let us look at the following example which deals with museum admissions at, let us say, the Museum of Natural History in New York City. Suppose that in one day, this museum collected $1590 from 321 people admitted to see its splendors. The price of each adult admission is $6. People between the ages of 4-17 pay the child admission of $4. Let us calculate how many of each type, adult and child, were admitted to the museum.

Linear systems give us a neat way of solving this problem. To appreciate the power of linear systems and the method of substitution, which we are going to use to solve this problem,
try to hazard a guess as to how you would figure this out. You will quickly see that there is no convenient way to get the number of adults and the number of children that were admitted. Yet by creating some linear models in the form of a system, we can quickly arrive at the answer to this problem.

Let us start with a verbal model and then translate this into mathematics. This is a convenient and helpful strategy which will allow us to solve the problem more readily. We have from the information that the number of adults plus the number of children is equal to 321. We also have that the number of adults times the price of an adult admission plus the number of children times the price of a child's admission is equal to the total amount collected.

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 price for adults is $6 and the price for children $4 and the total number of people attending 321, we have
x + y = 321 and 6x + 4y = 1590. Notice that both of these equations are in standard form. We can easily take the first equation and put into slope-intercept form by writing
y = -x + 321 (moving x to the other side). We now substitute into the second equation to get
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 since all the children were so good in doing something educational with their parents, they all went home with very nice souvenirs. Now isn't that a nice way to spend the day!

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