Consistent, Inconsistent, and Dependent Systems

A consistent system has two line that intersect in just one point, which means there is just one ordered pair that will satisfy both equations. An inconsistent system contains lines that do not cross, which means there are no ordered pairs that can satisfy either equation. A dependent system has two lines that intersect each other an infinite amount of times, giving it an infinite number of solutions.

Consistent, Inconsistent, and Dependent Systems

Let’s start with consistent systems. Here we have a consistent system, we have two lines that cross or intersect in exactly one point, that means there’s exactly one solution to a consistent system. That means there’s only one ordered pair that will satisfy both equations. If we plugged in 3 for x, and 2 for y, into both equations it would work in both.

That’s the only ordered pair that would satisfy both of these equations, since that’s the only ordered pair they have in common. Inconsistent systems. Notice that these lines don’t cross, and they never will, because they’re parallel lines. That means that they don’t have any ordered pairs in common, which means there is not a single point that would satisfy both equations.

Therefore, there is no solution to an inconsistent system. The last system is a dependent system. It looks like there’s only one line here, but really there are two, it’s just that they are the same line, so they’re on top of each other. That means that they cross each other or intersect each other an infinite number of times, which means there are infinitely many solutions.

Any ordered pair that would work for this equation, would also work for our other equation. You don’t have to graph these systems in order to tell if they’re consistent, inconsistent, or dependent, you can tell just by looking at the equations.

Looking at our system of equations for our consistent system, you can see that the slopes are different—and the slopes will always be different for consistent systems—so the slopes are not equal for a consistent system. If you have a system of equations with different slopes, then there’s only one solution to that system.

For an inconsistent system, notice our slopes are the same, but the y-intercepts are different, and that’s because these are parallel lines. The slopes are equal, but the y-intercepts are not. For the last system, the dependent system, it’s not as easy to tell looking at our equations because one is in standard form and the other is in slope intercept form.

However, if this equation in standard form was converted to slope intercept form, then the slopes would be equal, and the y-intercepts would be equal because these are the same lines, which means they must have equal slopes and equal y-intercepts.

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by Mometrix Test Preparation | Last Updated: June 13, 2019