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

In the x-y plane, line L passes through points (-1,-1) and (3,k). What is the value of k?

1) The y-intercept of line L is 1
2) The slope of line L is 2

\[{\text{2}}\,\,{\text{points}}\,\,{\text{of}}\,\,{\text{line}}\,\,L\,\,:\,\,\left( { - 1, - 1} \right)\,\,,\,\,\left( {3,k} \right)\]
\[? = k\]

\[\left( 1 \right)\,\,\left\{ \begin{gathered}
\,\left( {0,1} \right) \in L \hfill \\
\,\left( { - 1, - 1} \right) \in L \hfill \\
\end{gathered} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,L\,\,\left( {{\text{non - vertical}}} \right)\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,k\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}{\text{.}}\]

\[\left( 2 \right)\,\,\left\{ \begin{gathered}
\,\left( { - 1, - 1} \right) \in L \hfill \\
\,{\text{slop}}{{\text{e}}_L} = 2 \hfill \\
\end{gathered} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,L\,\,\left( {{\text{non - vertical}}} \right)\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,k\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}{\text{.}}\]

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

Regards,
Fabio.

=>

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 (k) and 0 equations in the original condition, D is most likely to be the answer. So, we should consider each condition on its own first.


We consider the equation of the line l, y = mx + b. Since it passes through the points (-1,-1) and (3,k), we can plug these points into its equation to yield -1 = -m + b and k = 3m + b.

Since we have 3 variables and 2 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since the y-intercept of line l is 1, we have b = 1 and m = b + 1 = 2.
Thus, k = 3m + b = 3*2 + 1 = 7.
Condition 1) is sufficient.

Condition 2)
Since the slope of line l is 2, we have m = 2 and b = m - 1 = 1.
Thus, k = 3m + b = 3*2 + 1 = 7.
Condition 2) is sufficient.

Answer: D

Note: When we checked the two conditions, we showed that both were equivalent in terms of b. So, each condition is sufficient by Tip 1)
of the VA method, which states that D is most likely to be the answer if conditions 1) and 2) provide the same information.

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.