SOLUTION: How do you know when an equation has infinitely many solutions?
How do you know when an equation has no solution?
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Question 314714: How do you know when an equation has infinitely many solutions?
How do you know when an equation has no solution?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
In the first place I really don't think that you are using the correct terminology. There are equations that are identities which could be construed to have infinitely many solutions, but I'm reasonably certain that is not what you refer to here. I believe you really want to know how to determine whether a system of equations has infinitely many solutions, which is to say that the system is consistent and dependent.
Generally speaking, when the process of solving a system of equations results in a triviality, such as 
, then you have a consistent and dependent system.
On the other hand, if the process results in an absurdity, such as 
, then the system is inconsistent.
When you are just working with a pair of 2-variable linear equations, sometimes you can tell just by looking at the two equations. It helps if they are both in the same form, such as standard form or slope-intercept form.
Given a system of 2-variable linear equations:
If 
, 
, and 
, then you have a consistent dependent system.
If and but 
, then you have a inconsistent system.
Given a friendly value of 
, you can tell by inspection.
This idea is extensible to systems of any number of dimensions. If all of the coefficients AND the constant term are proportional, then you are consistent and dependent. If all of the coefficients are proportional but the constant terms are not, then inconsistent.
John
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