Question 176056: {y=(x-2)^2+3
{y=|x-2|+3
i THINK that the system is dependent and consistant. Is that right?
thanks for your help!!
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! Not quite correct.
A consistent system has at least 1 solution.
An inconsistent system has no solutions.
A dependent system has an infinite number of solutions.
An independent system is uniquely determined, that is there is exactly 1 solution.
 means:
 →  for all  and  →  for all  ,
So: for . Substitute in  →  →  →  →  →  or 
→  and → 
Also: for . Substitute in →  →  →  →  → or  .
→ and →
Hence there are three ordered pairs that satisfy the system:
) and
) and
)
Now let's go back and review the definitions:
A consistent system has at least 1 solution. Yes, this system is consistent because it has at least 1 solution
An inconsistent system has no solutions. No, this system is not inconsistent because it has at least 1 solution
A dependent system has an infinite number of solutions. No, this system is not dependent because it has only three solutions, not an infinite number of solutions. A dependent system also has the characteristic that any ordered pair that satisfies one equation must also satisfy the other. In this case (0,5) would satisfy the absolute value equation, but not the quadratic.
An independent system is uniquely determined, that is, there is exactly 1 solution. No, this system is not independent because there is more than one solution.
See graphs of the two functions illustrating the three points of intersection:
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