Ellen S. answered • 04/08/14
Tutor
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Math and Writing Geek
Without specifically giving you the answers, let me see if I can point you in the right direction. When the question talks about the "solutions," it's referring to the value or values of x. a and b are constants, so they would in reality be replaced with numbers. The question is asking you what you can tell us about those numbers given a few different solution scenarios. Let's look at each one in turn.
If there are infinitely many solutions, that means that you should be able to substitute any number for x and have it still work out to be true. Can you think of the type of numbers you'd have to make a and b in order for that to be the case? Think about ways you could make that statement always be true no matter what the x value is. Hint: it'll probably involve the x being completely superfluous; a value that could be cancelled or factored out of the finished equation entirely.
If there is no solution, that means that when you try to solve it you should end up with an inconsistency such as 3 = 4, something that is blatantly untrue. How would you get the equation to come out to an inconsistency? Hint: this one probably also involves getting the x's to cancel or factor out completely, since you'll want to get to a point where you've only got constants and they don't work out right.
And if there is exactly one solution, that means you have to be able to solve it for x. What does that say about a? Is there a number that a cannot be if you want it to solve out correctly? Hint: This will feel like the opposite of the first two versions, in a way.
Hope that helps!
