Linear equations or functions are some of the more basic ones studied in algebra and basic mathematics. The import of these functions is that they model many real world phenomena and a key component of them, the slope, is a springboard concept for the realm of the calculus. That’s right: the basic idea of rise over run, or slope, within these equations, leads to all kinds of interesting mathematics.
A linear equation, or function, is simply one of the form Ax + By = C. The x and y are variables and the a,b, and c represent numbers like 1, 2, or 3. Usually the early letters in the alphabet represent numbers, or fixed, quantities and the latter letters in the alphabet stand for variables, or changing quantities. We use the words equation or function interchangeably, although there is a slight difference in meaning. At any rate, the expression Ax + By = C is known as a linear equation in standard form. When we move these expressions around and solve for y, we can write this equation as y = -A/Bx + C. When we substitute m for -A/B and b for C, we obtain y = mx + b. This latter representation is known as slope-intercept form.
The simplicity and utility of this form makes it special in its own right. You see, when a linear equation is written in this form, not only do we have all the information about the line that we need, but also, we can quickly and accurately sketch the graph. Slope-intercept form, as the name implies, gives us the slope, or inclination, of the line, and the y-intercept, or point at which the graph crosses the y-axis.
For example, in the equation y = 2x + 5, we immediately see that the slope, m, is 2, and the y-intercept is 5. What this means graphically is that the line rises 2 units for every 1 unit that it runs; this information comes from the slope of 2, which can be written as 2/1. From the y-intercept of 5, we have a starting point on the graph. We locate the y-intercept at (0,5) on the Cartesian coordinate plane, or graph. Since two points determine a line, we go from (0,5) up 2 units and then to the right 1 unit. Thus we have our line. To make our line somewhat longer so that we can draw its picture more easily, we might want to continue from the second point and go 2 more units up and 1 unit over. We can do this as many times as necessary to produce the picture of our line.
Linear functions model many real world phenomena. A simple example would be the following: Suppose you are a waitress at the local diner. You earn a fixed $20 per 8-hour shift and the rest of your income comes in the form of tips. After working at this job for six months, you have figured that your average tip income is $10 per hour. Your income can be modeled by the linear equation y = 10x + 20, where x represents hours and y represents income. Thus for the 8-hour day, you can expect to earn y = 10(8) + 20 or $100. You can also graph this equation on a coordinate grid using the slope of 10 and y-intercept of 20. You can then observe at any point in your day where your income stands.
Simple models like these show us how mathematics is used in the world around us. Having read and digested the contents of this article, try to come up with your own example of a linear equation or model. Who knows? You just might start liking math more than ever before.