Is x = y? (1) x^2 = xy (2) x^2 = y^2

Is \(x=y\)?

(1) \(x^2=xy\) --> \(x^2-xy=0\) --> \(x(x-y)=0\) --> either \(x=0\) (and \(y=any \ value\)) OR \(x=y\). Not sufficient.

(2) \(x^2=y^2\) --> either \(x=y\) or \(x=-y\). Not sufficient.

(1)+(2) \(x^2=xy=y^2\) --> \(x=y\). Sufficient.

Answer: C.


When you cancel \(x\), what you are actually doing is dividing both parts of the equation by variable \(x\), thus assuming with no ground for it that this variable does not equal to zero.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.


Is the product of A and B equal to 1?

Question: \(ab=1\)?

(1) \(a^2b=a\) --> \(a^2b-a=0\) --> \(a(ab-1)=0\) --> either \(a=0\) (and \(b=any \ value\), including zero) so in this case \(ab=0\neq{1}\) OR \(ab=1\). Two different answers, not sufficient.

(2) \(ab^2=b\) --> \(ab^2-b=0\) --> \(b(ab-1)=0\) --> either \(b=0\) (and \(a=any \ value\), including zero) so in this case \(ab=0\neq{1}\) OR \(ab=1\). Two different answers, not sufficient.

(1)+(2) either \(a=b=0\), so in this case \(ab=0\neq{1}\) and the answer to the question is NO, OR \(ab=1\) and the answer to the question is YES. Two different answers, not sufficient.

Answer: E.

When we consider two statements together we should take the values which satisfy both statements. For this question \(ab=1\) satisfies both statement, but \(a=b=0\) also satisfies both statements. So what you call "common solution" for this question is: \(ab=1\) OR \(ab=0\neq{1}\) and for the first question "common solution" is only \(x=y\).

Hope it's clear.