Select for a and for b values such that the equation x/a = x - b has

Question

Select for a and for b values such that the equation \(\frac{x}{a} = x - b\) has more than one solution for x. Make only two selections, one in each column.

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Bunuel · Accepted Answer

Notice that x/a = x - b represents a linear equation with one unknown, x. Generally, such equations have one solution for x. For instance, x/2 = x - 1, which has one solution: x = 2. Similarly, x/4 = x + 3 also has one solution: x = -4. The only way for x/a = x - b to have more solutions is if the left-hand side and the right-hand side are identical, resulting in x = x, which is true for all values of x, hence more than one solution. This could happen if a = 1 and b = 0, thereby leading to x = x, and infinitely many solutions for x. https://gmatclub.com/forum/download/file.php?id=81992 GMAT-Club-Forum-9k18wenv.png GMAT-Club-Forum-1eu7xuzj.png

chetan2u · Answer

\frac{x}{a} = x - b 
 x=ax-ab......ax-x=ab.......(a-1)x=ab

Now, when will x have more than one value, when we have 0*x=0.  
Here x can be 1, 2, 3, 0.5 and so on. Rather, the equation will be satisfied irrespective of the value of x.

So, a-1 =0 or a=1.
Also, ab=1*b=b=0 or b=0

Answer: a=1 and b=0

a_b_p · Answer

if equation is further simplified we get, x = ab/(a-1). => a cannot be 1. Seems to me that the question is wrong.

chetan2u · Answer

You are dividing ab by 0, which is not correct. When a-1 = 0, you cannot divide ab by it.

(a-1)x=ab is perfectly fine.

Say 0*x = b.....
You cannot now take 0 on other side, that is x=b/0, and say 0*x=b i snot valid.

Pr4n · Answer

I am not a math expert, but that is true when the single variable eqn has only 1 solution. It has more than 1 and it is not mentioned that it has 2 or 3, it could have unlimited solution if you think about it. So actually when denominator is 0, the result is undefined. And in a way when we create the situation x=x, x here is undefined as you can't pick any  1  value for x, this is true for all x's. Plus focus on the original equation maybe after simplifying yes it shouldn't take 1 as a value, but the original equation doesn't restrict it anyway to take that value.

Also imagine if b was 0 indeed then the eqn becomes x= ax, which can only be true if a =1. So if you take any other value of a then also the eqn itself breaks. And i could agree this question mathematically might seem illogical at first glance, but from a reasoning perspective and what the question is asking it is indeed correct.

monarchme · Answer

One way to solve this that when you rearrange the variables, you will get 
x= 
\frac{-ab}{1-a}

now we know that on normal days we get one value for x, and that is called its " deterministic " solution. A solution which one can   determine.   We also have " indeterministic " forms, those forms in which we cant determine a single value, examples are:0/0, ∞/∞, 0 × ∞, ∞ - ∞, 00, 1^∞, and ∞0.

In this question we willingly have to make  \frac{-ab}{1-a}  into an  indeterministic  form so that we have two solutions.
by putting a=1 and b=0, we have 0/0 and thus the answer.

KarishmaB · Answer

Responding to a pm:

We have a linear equation in x (an and b are all constants as we can see from the options). Hence whatever the values we give to a and b, we will automatically get a single unique value for x. There is no way to introduce an x^2 or x^3 etc to get 2 or 3 values for x. The only other option could be infinite values i.e. an equation that holds for all values of x. 
Then if b = 0 and a = 1, we get x = x, an equation that holds for EVERY real value of x.

Hence  a = 1, b = 0

Select for a and for b values such that the equation x/a = x - b has

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Select for a and for b values such that the equation x/a = x - b has