SOLUTION: How do you determine consistency from using the elimination method?

Question 130788: How do you determine consistency from using the elimination method?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Consistency means the system has at least one solution. The three possibilities for any system of equations are:
1. No solution, meaning that the equations graph to parallel lines so that they never intersect. Using the elimination method on such a system will result in a clearly false statement such as 0 = 5.

Example:

Multiply equation 1 by -2 and then add the equations:

. This is clearly a false statement, so there is no solution and the system is inconsistent.

2. Infinite solutions, meaning that the equations graph to the same line. In other words ANY ordered pair that satisfies one of the equations will satisfy the other. Using the elimination method on such a system will result in a trivial identity such as 5 = 5.

Example:

Multiply equation 1 by -2 and then add the equations:

. This is a trivial identity statement, therefore any ordered pair that satisfies one of the equations will satisfy the other, and there are an infinite number of such ordered pairs. This system is consistent, but it is not uniquely determined (also refered to as dependent)

3. One solution, meaning that the equations graph to two different lines with differing slopes and that intersect in exactly one point. The ordered pair that represents the point of intersection is the only ordered pair that satisfies both equations.

Example:

Multiply equation 1 by -2 and then add the equations:

=> => (0,10) is the single point of intersection of the two lines represented by the two equations.

This is a single ordered pair and the only ordered pair that satisfies both equations. This system, because it has at least one solution, is consistent. Because it has only one solution, it is uniquely determined (also known as independent)

Review: When applying the elimination method, if you get:

1. A statement that is always false, the systems are inconsistent.

2. A statement that is trivially true, the systems are consistent but not uniquely determined (or dependent).

3. A single value for one of the variables, the systems are consistent and uniquely determined (or independent).
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