Section 2.5 Graphs of Linear Equations

Consider the linear equation \(8x+4y=12\text{.}\) Before we try to graph this equation, let’s generate a table of ordered pairs for this relation. To do so, we can pick any \(x\) values and find their corresponding \(y\) values, or vice versa. Some ordered pairs are shown below.

We can now plot these points. All of these points fall in a straight line; connecting the points, we get the graph of the equation \(8x+4y=12\text{.}\) We add arrows on either end to indicate that the graph extends indefinitely in both directions.

Let’s create a table and graph the linear equation \(2y-x=8\text{.}\) Again, we start by generating a table with several ordered pairs that satisfy this equation.

Notice that several of the ordered pairs in our table have decimal-valued coordinates. If you are randomly selecting values of \(x\text{,}\) this will often happen and is okay. However, if you are plotting these points by hand and cannot accurately plot decimal-valued points, you should avoid using these points. In the table above, we found two points with integer coordinates: \((0,4)\) and \((2,5)\text{.}\) These two points are enough to graph this equation. We can then verify that all of the decimal-valued coordinates do in fact fall on this line.

It is often useful to describe where the graph of a linear equation meets the \(x\)- and \(y\)-axes. The point where a graph intersects the \(x\)-axis is called the \(x\)-intercept of the graph, and the point where a graph intersects the \(y\)-axis it called the \(y\)-intercept.

It is not always clear from the graph what the intercepts of a line are. For example, consider the graph of the equation \(6y-3x=-17\text{.}\)

It is not clear where exactly the line is intersecting the axes. Although we could estimate where these points lie, there is a more accurate way to find the intercepts.