The Meaning of Graphs

In my last post, I showed how to plot points on a coordinate system. It is important to remember that a point such as (2, 1) means that for that point, x = 2 and y = 1. For the point (-5, -3), x = -5 and y = -3. The first number is always the x value and the second point is the y value. With that as a background, let’s talk about how to graph an equation.

Now we can solve an equation with one unknown like


Now past posts have talked about how to formally solve an equation like this but I think you can readily see that the solution to this equation is x = 4. That is, if you replace the x with 4, you get a true statement that 4 + 3 = 7. Now this is one equation with one unknown and there is only one solution. But what about y = x + 3 ? Here there are two unknowns, y and x. But you can come up with several solutions. If x = 4, then y = 7. If x = 5, then y = 8. If x = 1, then y = 4. If x = -2, then y = 1. You can see that there are many solutions to this one equation with two unknowns. In fact, there are an infinite number of solutions especially when you consider that fractional numbers are allowed as well. For example, if x = 2.67, then y = 5.67.

Now you see that the solutions are pairs of numbers: an x and a y. So we can think of a solution as a point on a graph. Since any point plotted on a coordinate system is of the form, (x, y), two of the solutions can be shown as (4, 7) and (-2, 1). If I plot these points and others that are solutions to the equation, it appears that all the points that make this equation true are on a line. in fact, they are and I’ve drawn the line over the points:

The main point here is that the graph of this equation is a picture of all the (xy) pairs that satisfy the equation. By looking at this, you can pick out other solutions like (3, 6) or (-3, 0). All the fractional points that satisfy this equation are also on the line. Since there are an infinite number of points that satisfy this equation, the graph is a solid line. Any graph, even curvy ones, are a picture of all the (xy) pairs that satisfy the equation that generated the graph.