How to Graph Consistent, Inconsistent & Dependent Linear Equations

Instructor: David Karsner

David holds a Master of Arts in Education

When two linear equations are graphed on the x,y coordinate grid, you have three possible results. The two equations can be consistent, inconsistent, or dependent. This lesson explains how to graph these equations and what the graphs mean.

Runners Take Your Mark

Josh and Will are running three 500 meter races against each other. Each race has a different outcome.

Race 1: Will has the 20 meter head-start, but Josh runs 1 meter per second faster than Will. Josh ends up passing Will after 20 seconds and wins the race.

Race 2: This time Josh gives Will a 20 meter head-start. They both run a consistent 7 meters per second. The entire race Will is exactly 20 meters ahead of Josh. Josh never gets any closer.

Race 3: Josh and Will both start the race at the zero meter mark and run 7 meters per second for the entirety of the 500 meters. They stay right beside each for the duration of the race and end up in a tie.

If you were to graph these runners on a grid with seconds on the x-axis and meters on the y-axis, Race 1, 2, and 3 would show you the three different possibilities when graphing linear equations. Race 1 demonstrates a consistent graphing. Race 2 shows a inconsistent graphing. Race 3 is the dependent graphing. This lesson will define the different possibilities and instruct you on how to graph them.

The Graph of Linear Equations

The graph of a linear equation will always be a line, hence the name 'linear.' When you graph two equations onto to the same x,y coordinate grid, you only have three possibilities. One possibility is that the lines will intersect at one point. A second possibility is that they never intersect and the final possibility is that they have all points in common; they intersect at every point.

There are several ways to graph a linear equation. Since the graphing of all three possibilities depends on knowing the slope and y-intercept; putting the equations in the y=mx+b form would be the simplest approach. You will need to move the parts of the equation until you have y isolated on one side of the equation. If the equation is in the form of y=mx+b; then the point (0,b) is the y-intercept and the m is the slope, which tells you how to move from the y-intercept when graphing.

For example, if you were to graph the equation y=3x+5 you would graph the point (0,5) and move up 3 and to the right one spot and plot another point. The line would go through both of those points.

Consistent Equations

We mentioned in the last paragraph that one possibility for the two lines is for them to meet at only one point. If that is the case, then these two lines are consistent equations. Consistent equations will intersect at only one point. The graph of these two equations will look like an X. There is only one point, the point of intersection, that will make both equations true. If you put the x and y from the point of intersection into both equations, then the equality will hold for both.

If two lines meet at only one point they will have different values for their slopes. You should put both equations in the y=mx+b form. If the two equations have different values for m then the two lines will intersect at exactly one point.

Consistent Linear Equations

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