Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

In Definition 4.4, we saw that solving an equation means to find the value (or values) of the variable (or variables) that make the equation true, and that solutions are the specific values of the variables that accomplish that. We are going to take another look at those ideas, but in the context of using multiple variables simultaneously. Specifically, we are going to be working with linear equations in the variables \(x\) and \(y\text{.}\)

Long lists of ordered pairs are not the most intuitive way to present solutions. For more than just a couple points, it makes sense to transition to using a chart, such as the one in the margin. Sometimes a chart like this can help us see a pattern in the numbers if the pattern is simple. But even with that, charts have limited value because it’s still just a list of numbers. It would be better to use a more visual representation of this information.

Mathematicians often use a coordinate plane to represent solutions to two variable equations. The coordinate plane is a picture where specific positions represent specific ordered pairs. Most students are familiar with the basic design of the standard rectangular coordinate grid, which is the grid we will be using.

Students often learn coordinates as a sequence of motions starting from the origin. The point \((2,3)\) is located by starting from the origin, moving right 2, and then moving up 3. Negative \(x\)-coordinates correspond to moving to the left instead of to the right, and negative \(y\)-coordinates correspond to moving down instead of up.

But there is another way to look at this which is a little bit more general. Rather than thinking about this in terms of movement, we can think about this in terms of the intersection of two lines. Lines of the form \(x = \text{(Number)}\) are vertical lines corresponding to the coordinates on the \(x\)-axis and lines of the form \(y = \text{(Number)}\) are horizontal lines corresponding to the coordinates on the \(y\)-axis. It is the overlap of these lines that creates the coordinate grid.

Once you have this, then you can see that the positions are actually the intersection of two of these lines, corresponding to the specific value and the specific value. Here are the two ways of interpreting the point \((x,y) = (2,3)\) visualized side-by-side.

We can generalize the idea of giving locations as the intersection of two lines by allowing ourselves to use curves. For example, locations on the earth are found as the intersection of the latitude and longitude lines (which are actually curves on the globe). Your location in a city can often be described as being near the intersection of two streets (which may not be straight). In trigonometry, there’s another coordinate grid that’s built around circles and lines pointing out from the origin called polar coordinates.

Once we have the ability to locate points on the coordinate plane, we can then plot lots of points on the same coordinate grid and look for a pattern.

An important feature to recognize is that the line that is drawn represents all of the solutions. It turns out that every single point on the line will solve the equation, even the ones that fall in between the grid points. This is where all those decimal solutions can be found.

However, while the sketch of the line is an important tool for building intuition, you need to be very careful about trying to guess exact values on the basis of a sketch. If you’re reading values from a graph that are not on the grid lines, you must always acknowledge that you are only giving an approximation.

PDF Version of these Worksheets

¹

external/worksheets/12-Worksheets.pdf

Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.

This section focused more on developing your intuition for lines than on the algebra. Linear equations are extremely common in practical applications. For example, if a burger costs $3 and you want 5 burgers to bring to your friends, how much will that cost? This can be modeled as a linear equation.

Of course, simple cases such as this can be done with simple methods. As you start to work with "real life" situations, it’s usually not going to solved by quick mental arithmetic. You’re going to need better tools in your toolbox. And this is where having a background in algebraic reasoning and mathematical thinking will help.

As you worked your way through the worksheets, you might have started to notice that moving from one point to another along a line always required your variables to change in tandem. For example, the might increase by every time the decreased by We will see that this is a reflection of the idea of the "slope" of a line, which we will explore more deeply in the next section.

Without telling you, there were two types of modeling that were introduced in the last couple worksheets. These are known as interpolation and extrapolation. The basic idea of interpolation means to fill in data points between existing data points. (See Section 12.5.) Extrapolation means modeling data beyond the existing data points. Linear equations are often used for models of these types. It’s usually more complicated because real life data rarely ever fits exactly on a line, so you have to go through a process of finding the "best" line that matches the data, but that application will have to wait for a science or business course.

Most of the time in math courses, we use the variable for the horizontal axis and for the vertical axis. This is the "standard" choice for coordinate systems when there is no specific context being applied. But a lot of mathematical ideas are applied in very specific contexts, and so we need to be able to translate our ideas from the generic situation to specific ones.

Traditionally, the \(x\) variable represents the input (or independent) variable, and \(y\) represents the output (or dependent) variable. This language is tied to how mathematicians talk about functions. We will start with the formal definition:

Conceptually, functions are often thought of as machines. On one side of the machine there’s a place for you to put something in, and on the other side there’s a place where something comes out.

Specifically what comes out will depend on what the machine is designed to do (the "rule" that it has been created to follow). For example, we can have an "Add 3" machine that takes whatever number we give it and add 3 to it.

If we picked a different input, we would get a different output.

In fact, if we were to give it a generic quantity, it could give us an expression that represents the appropriate output.

The mathematical shorthand for this concept is \(y = f(x)\text{.}\) This is a versatile notation that we use both to describe the function in general as well as to describe specific input-output pairs of a specific function. And while we often use the letter to represent a function, we can use other letters or symbols if we wanted. The key observation about this notation is that it defines as the independent variable and as the dependent variable. For the "Add 3" example, we would write \(y = f(x) = x + 3\text{.}\)

Notice that this is a process that gives us ordered pairs \((x,y)\text{.}\) If we were to put these values into a chart like the ones we were working with earlier in this section, we could convert this information into a graph.

There are situations where the data that you have in your table comes from measurements. For example, if you’re measuring the temperature throughout the day, you will get a chart of values, but the values probably won’t fit a nice formula. Even though there may not be a formula, there’s still a meaningful sense that this is a function. The input variable is the time and the output variable is the measurement. And using the same ideas as above, we can put that information into a chart and get a graph of the temperature throughout the day.

Notice that we had to make some modifications to the graph. We changed the letters to better reflect what we’re measuring (\(t\) for time and \(F\) for degrees Fahrenheit), and we also labeled the axes so that the values could be more easily interpreted. We also altered the time axis to correspond to clock times instead of just having numbers.

But notice how natural this is to read. Given a time, we can get an estimate of the temperature by looking at the graph. This is where a lot of the value of graphing functions starts to come into play. There is a clear relationship between the input value (time) and the output value (temperature). Even if we picked a time that isn’t one of the data points, we can still come up with an estimate for the temperature.

You might have noticed that we connected the data points together with straight lines, and that makes the graph look a little artificial. That’s almost certainly not how the temperature behaved! It’s far more likely to have a smooth shape. This is also why we said that we could estimate the temperature instead of saying that we could determine the temperature. There’s a little bit of guesswork involved in thinking through the shape of the curve in between the data points.