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by Jay@ManhattanReview » Wed Aug 30, 2017 8:18 pm
danielle07 wrote:Are the lines with equations 2x + y = 2 and x - 2y = 0 parallel, perpendicular or neither?
Hi danielle07,

Pl. post the complete question with five options. The narration suggests that it is not a GMAT question; however, the concept tested is within the scope of the GMAT.

Note that the following for two straight.

1. y = mx + c; where m = magnitude (a measure of the slope of a line) of a line, and c = y-intercept of the line
2. y = m'x + d; where m' = magnitude of the line, and d = y-intercept of the line

a. The lines are perpendicular if the product of the magnitudes (a measure of the slope of a line) of the lines equals -1.

=> m x m' = -1

b. The lines are parallel if the magnitudes of the lines are equal.

=> m = m'

Let's come back to the question.

We have two equations of straight lines.

2x + y = 2 ---(1)
x - 2y = 0 ---(2)

Let's transform the equations to the standard y = mx + c form.

Thus,
2x + y = 2 => y = -2x + 2 ---(1); thus, m = -2
x - 2y = 0 => y = (1/2)*x ---(2); thus, m' = 1/2

Since m ≠ m', the lines are not parallel.

Let's find out the value of m x m'. m x m'= -2 x (1/2) = -1.

Thus, the lines are parallel.

Hope this helps!

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-Jay
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by Matt@VeritasPrep » Thu Aug 31, 2017 4:31 pm
Get the slopes in the same form, y = mx + b:

y = -2x + 2
and
y = (1/2)x

Two lines are perpendicular if the product of their slopes is -1.

Since the slopes are -2 and 1/2, respectively, the lines must be perpendicular: -2 * (1/2) = -1.
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Answer

by Admin1 » Fri Sep 01, 2017 7:43 pm
Let's remember this: given a line with the equation Ax+By+C=0, the slope of the line is m=-A/B. On the other hand, given two lines: L1: A1x+B1y+C1=0 and L2: A2x+B2y+C2=0,
- L1 and L2 are parallel if their slopes are the same and C1 is not equal to C2,
- L1 and L2 are the same line if they are parallel and C1=C2.
- perpendicular if the product of their slopes are equal to -1,
- secants (they intersect each other) if neither option above is satisfied.

So the slopes of the given lines are:
m1= -2/1=-2
and
m2=-1/-2=1/2

As m1 is not equal to m2, the lines are NOT parallel, but we can see that: m1*m2=-2*(1/2)=-1, so the lines are perpendicular.
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