Graphing Summary
Once you’re able to solve equations that have one variable, you can start to look at equations that have two variables. Typically we like to use the variables x and y for this purpose. So suppose we’re given the equation y = 2x + 1 and we want to solve it.
As it turns out, there will be a large number of solutions for this type of equation instead of just one. We can write these solutions in ordered pairs. An ordered pair of the form (x, y) could be something like (2, 5). To see if the ordered pair (2, 5) is a solution for the equation y = 2x + 1, then we just plug in 2 for x and 5 for y and see if we get something that makes sense:
y = 2x + 1
5 = 2(2) + 1
5 = 4 + 1
5 = 5
So we see that (2, 5) is a solution for the equation y = 2x + 1. We can find other solutions by plugging in different values for x and solving for y. For example, let’s plug in 1 for x and figure out what the corresponding y value will be:
y = 2x + 1
y = 2(1) + 1
y = 2 + 1
y = 3
This shows us that when x is 1, y is 3. The corresponding ordered pair would be (1, 3). We could do a few of these in a row, and find many different ordered pairs. If we solve for y when x = 1, when x = 2, when x = 3, and when x = 4, we’ll get the ordered pairs (1, 3), (2, 5), (3, 7), and (4, 9). If we plot these four points on a graph, we’ll notice something interesting:
On this graph above, in red are the four ordered pairs we found. What you’ll notice is that they are all in a line, which is marked in blue. What we’re showing you here is that this happens because the equation y = 2x + 1 is, in fact, the equation of a line shown in blue.
By studying the nature of equations like y = 2x + 1, y = -3x + 4 or y = -x + 9, we can start to see some patterns and learn a lot more about graphing these lines.
Slope – How steep a line is compared to the x-axis and y-axis is one of the two important measures of a line.
Slope-Intercept Form – Using this basic format for the equation of a line makes it easy to graph and understand.
Graphing Lines – Taking the equation of a line and graphing teaches you a lot about graphs and the nature of linear equations.