Data Sufficiency

[Math Revolution GMAT math practice question]

For integers m and n, the operation $$\triangle$$ is defined by
$$m\ \triangle\ n\ =\ \left(m-1\right)^2+\left(n+1\right)^2$$
What is the value of the integer x?

$$\left(1\right)\ x\triangle1\ =4$$
$$\left(2\right)\ 1\triangle x=4$$

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

For any integers m and n,
$$m\ \triangle\ n\ =\ \left(m-1\right)^2+\left(n+1\right)^2$$
What is the value of the integer x?

$$\left(1\right)\ x\triangle1\ =4$$
$$\left(2\right)\ 1\triangle x=4$$

$$m\ \triangle\ n\ =\ \left(m-1\right)^2+\left(n+1\right)^2 \,\,\,\,\,\left[ {m,n\,\,{\rm{ints}}} \right]$$
$$? = x\,\,\,\,\,\,\left[ {\,x\,\,{\mathop{\rm int}} \,} \right]$$
$$\left( 1 \right)\,\,\,4 = x\,\vartriangle \:1\,\, = \,\,{\left( {x - 1} \right)^2} + {\left( {1 + 1} \right)^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\left( {x - 1} \right)^2} = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,x = 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{SUFF}}.\:$$
$$\left( 2 \right)\,\,\,4 = 1\,\vartriangle \: x\,\, = \,\,{\left( {1 - 1} \right)^2} + {\left( {x + 1} \right)^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\left( {x + 1} \right)^2} = 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,x = 1\,\,\,{\text{or}}\,\,\,x = - 3\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\text{INSUFF}}.\:$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
$$x\ \triangle\ 1\ =\left(x-1\right)^2+\left(1+1\right)^2=\left(x-1\right)^2+2^2=4$$
Thus, (x-1)^2 = 0 and x = 1.
Since we have a unique solution, condition 1) is sufficient.

Condition 2)
$$1\ \triangle\ x=\left(1-1\right)^2+\left(x+1\right)^2=\left(x+1\right)^2=4$$
So, x+1 = ±2 or x = -1 ± 2.
Thus, x = -3 or x = 1.
Since we don't have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.