Problem Solving

[Math Revolution GMAT math practice question]

The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersection in the xy-plane. Which of the following could be the value of p?

A. 0
B. 1
C. 2
D. 3
E. 4

Max@Math Revolution wrote: The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersection in the xy-plane. Which of the following IS the value of p?

A. 0
B. 1
C. 2
D. 3
E. 4

\[? = p\]
From the question stem, we know both lines (each represented by one of the equations) must coincide (*), hence:
\[\left\{ \begin{gathered}
\,x + 2y = 3\,\,\,\left( { \cdot 2} \right) \hfill \\
2x + py = q \hfill \\
\end{gathered} \right.\,\,\,\,\,\, \sim \,\,\,\,\,\,\left\{ \begin{gathered}
\,2x + 4y = 6 \hfill \\
2x + py = q \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\,? = p = 4\,\,\,\,\,\,\,\left( {{\text{and}}\,\,q = 6} \right)\]

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

Regards,
fskilnik.

=>

If the 2 lines have infinitely many points of intersection, their equations must specify the same straight line.
The equation x+2y=3 is equivalent to 2x+4y=6.
So, p = 4 and q = 6.

Therefore, the answer is E.
Answer: E