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# Solutions to systems of equations: consistent vs. inconsistent

A consistent system of equations has at least one solution, and an inconsistent system has no solution. Watch an example of analyzing a system to see if it's consistent or inconsistent. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Does anyone know why they call it "consistent" or "inconsistent", not some other word?
• "Inconsistent" is because it is not possible for both equations to hold simultaneously. They contradict each other in the sense that if one holds, the other must fail. Thus their graphs never intersect and there is no solution to their system.

"Consistent" is then the opposite. There does exist solution(s) to the system.
• when you graph a system of equation can you have 2 solutions?
• Yes, if the system includes other degrees (exponents) of the variables, but if you are talking about a system of linear equations, the lines can either cross, run parallel or coincide because `line`ar equations represent lines.
If you are graphing a system with a quadratic and a linear equation, these will cross at either two points, one point or zero points.

If you have a quadratic like y = x² + 2x -3 and a linear equation like y = -x + 1 , this example intersects at two points, (-4,5) and (1, 0), so this system does have two solutions.

If you have a quadratic like y = x² - 2x +1 and a linear equation like y = 2x - 3, this example intersects at one point, x = 2. y = 1 so the point (2,1) is the only solution to this system of equations.

If you have a quadratic like y = x² - 2x + 1 and a linear equation like y = (1/5)x - 2
these never cross and so there will be no solution for this system of equations.
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A straight line and a quadratic will not coincide, because a quadratic equation represents a parabola--a very un-linear curve! They cannot match at every point.
• I really don't get how a consistent system can be overlapped
• Hi Isaiah,
When you say "overlapped" do you mean the lines crossing? That's essentially the definition of a consistent system - that there is a solution, which is that point where the lines cross.

Or when you say "overlapped", maybe you mean that the two lines are the same and they overlap each other from start to finish? That is considered a consistent system too. When your two equations graph to the same line, the solution is all the points on the line, a solution set, rather than just one point. The video "Infinite solutions to systems" has an example of that situation.
• Unless it is the same line, couldn't you just look at the slopes?
If the slope is the same, then it is inconsistent. (unless it's the same line)
If the slope is different, then they have to intersect.

• As Sal said in the video, you can do it. He just wanted to make it clearer to view.
• What do I do if there are 3 variables? Is there a way to figure this out by intuition, other than, of course, generating these vectors in my mind!?
• To solve a system with three variable you need three equations.

Combine them in two sets of two to get rid of one variable. Then combine these two equations to get rid of another variable.
• So a consistent line either has one solution or infinite solutions, right?
• It's not right to say "a consistent line." You need more than 1 line to have either a "consistent" or "inconsistent" system. Then you can say that a consistent system (with at least 2 lines) has one solution or infinite solutions.
• x-y=2 is it inconsistent?
• That is not a system of equations, so a single equation cannot be inconsistent, it is just a linear equation. If you had 2x - 2y = 14 as a second equation, then the two would be inconsistent.
• Technically speaking these equations are being mapped on a 2-dimensinal scale right? Is there such a thing as mapping a 3-dimensional equation, or even a 4-dimensional one? Also I would like to know how those would be used and why.
• Yes these are based in planar geometry. So imagine a square room with 4 walls, a floor, and a ceiling. You see all sorts of lines that could be parallel (if on the same plane which could include planes you do not see like the wall and ceiling on one side of the room and the wall and floor on the opposite side of the room). Similarly, there are a lot of intersecting lines at the corner of the room. Any two lines would still be on a single plane.
The difference in 3d would be that each corner is the intersection of 3 lines which are not on the same plane and there are some lines that are skew to each other, not on the same plane, but also not parallel like the wall and ceiling of one wall and an adjacent wall and floor. So if you have two rooms that are aligned, you could end up with the same like with one long wall divided by an internal wall. I do not know about 4d. They are used in architecture, game design, and anything that needs 3 dimensions.
• But what if we have a system of equations of a plane (ie. 3 variables)? Then how do we determine whether the planes are consistent or inconsistent?