Simultaneous Equations

In the video below, the Portuguese soccer star Cristiano Ronaldo is racing a car. I am left to wonder just how fixed this whole thing is, but let's pretend for a second that it isn't. In either event, it makes for some good math problems.

In particular, I would like to see if we could use math to determine whether it would ever be possible for Cristiano Ronaldo to actually win a race like this one against a Bugati. And, if so, how we should set the race up to make this happen. There are a number of important concepts from high school mathematics that are involved in solving such a problem, including understanding the behavior of polynomials as well as how to model a real world phenomena using systems of equations and the techniques to solve such equations.

To make things easier, let's start with a simpler case than the scenario we see in the video above. In particular, let us consider what happens if the equations that we are trying to solve are linear equations, and can be expressed as a line. An overview of the equation of a line, as well as a test to check your understanding, are given in the post Algebra v. Cockroaches.

THE EQUATION OF A LINE

One of the most repeated concepts in teaching high school algebra is the concept of linear equations. It is almost a little bit annoying how much they stress this concept without telling you why it is important. For instance, you may remember or already know that the equation of a line is given by the form . . .

y = mx +b

.. where we say that m is the slope, and b is the y-intercept. To understand what these terms mean, it is helpful to look at the graphs of some lines. The slope of the line is a measure of how steeply the line increases or decreases. In the picture below, line (1) has a positive slope because it is increasing as we move to the right. Line (2) has a negative slope because it is decreasing as we move to the right.

In a scenario like a race, or an object moving with some speed, a linear equation arises when that object is traveling at a constant speed. So let's say that we were driving a car at 55 miles per hour, and we wanted to know how far we drove in 3 hours. Or 2.5 hours, or 10 hours?

To arrive at an answer, we would multiply 55 mph by the amount of time spent traveling in hours, and that would tell us how far we have gone. In this case, we could write a linear equation where y represents the distance traveled in miles, and x represents the time traveled in hours. In this case, our equation would be

y = 55x

In this case, if we were to graph this equation, we would see that it was essentially plotting the position of the car at different times. And if we were to draw this graph, it would appear to be a line with a slope of 55. In this equation, the y-intercept, or b, is equal to zero.

Now let's say that instead of measuring how far we had driven, we wanted to know how far away we were from our house. If we started driving when we were 7 miles away from our house, and we were traveling at 30 mph, what is the equation we could use to determine how far away we are from our house?

In this case, our equation would be

y = 30x + 7

and this graph appears as well in

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