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The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio

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The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio  [#permalink]

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New post 27 Sep 2018, 00:59
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E

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[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 \(p\)?

\(A. 0\)
\(B. 1\)
\(C. 2\)
\(D. 3\)
\(E. 4\)

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Re: The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio  [#permalink]

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New post 27 Sep 2018, 03:03
If the two lines have an infinite number of intersections, then they must be overlapped. Their equations should then be the same.

Equation of line 1, isolating y: y=-x/2 + 3/2

Equation of line 2, isolating y: y=-2x/p + q/p

We want the same slope so p should be 4 in order to obtain y=-x/2 + q/4
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Re: The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio  [#permalink]

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New post 27 Sep 2018, 08:34
MathRevolution 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.
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Re: The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio  [#permalink]

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New post 27 Sep 2018, 10:26
Infinite intersection means they are on top of each other , which in turn means their slope is the same.
Option E
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Re: The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio  [#permalink]

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New post 30 Sep 2018, 22:56
=>

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
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Re: The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio &nbs [#permalink] 30 Sep 2018, 22:56
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