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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42 GPA: 3.82
The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio  [#permalink]

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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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1
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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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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Infinite intersection means they are on top of each other , which in turn means their slope is the same.
Option E
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8005
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The 2 lines x+2y=3, 2x+py=q have infinitely many points of intersectio  [#permalink]

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=>

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