Harshgmat wrote:
What is the value of a ?
a) \(a^2 = b^2\)
b) b - a = -12
To prove that each statement alone is insufficient, we present a BIFURCATION (algebraic, in this case):
\(? = a\)
\(\left( 1 \right)\,\,{a^2} = {b^2}\)
\(Take\,\,\left\{ \begin{gathered}
{a^2} = {b^2} = 0\,\,\,\,\, \Rightarrow \,\,\,\,? = 0 \hfill \\
{a^2} = {b^2} = 1\,\,\,\,\, \Rightarrow \,\,\,\,? \ne 0\,\,\,\,\,\,\,\,\left( {a = \pm \,1} \right) \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{INSUF}}{\text{.}}\)
\(\left( 2 \right)\,\,a\, - b = 12\)
\(Take\,\,\left\{ \begin{gathered}
b = 0\,\,\,\,\, \Rightarrow \,\,\,\,? = 12 \hfill \\
b = 1\,\,\,\, \Rightarrow \,\,\,\,? \ne 12\,\,\,\,\,\,\,\,\left( {a = \,13} \right) \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{INSUF}}{\text{.}}\)
\(\left( {1 + 2} \right)\,\,\,\,\,\left\{ \begin{gathered}
{a^2} - {b^2} = 0\,\,\,\,\, \hfill \\
a - b = 12 \hfill \\
\end{gathered} \right. \Rightarrow \,\,\,\,\,\,\left\{ \begin{gathered}
\left( {a + b} \right)\left( {a - b} \right) = 0 \hfill \\
a - b = 12 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \begin{gathered}
a + b = 0 \hfill \\
a - b = 12 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,2a = 12\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a\,\,{\text{unique}} \Rightarrow \,\,\,\,\,\,{\text{SUF}}{\text{.}}\,\)
The above follows the notations and rationale taught in the GMATH method.
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Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
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