SOLUTION: 6x+12y=24 X-y/2=18 Are these lines parallel or perpendicular?

Question 547769: 6x+12y=24
X-y/2=18
Are these lines parallel or perpendicular?
Found 2 solutions by mananth, lmeeks54:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
6x+12y=24
12y=-6x+24
/12
y=-1/2 x +2
X-y/2=18
y/2 = x-18
y=2x-36
slope of first line = -1/2
slope of second line = 2
-1/2 * 2= -1
so the lines are perpendicular
Answer by lmeeks54(111) (Show Source): You can put this solution on YOUR website!
To answer the question about parallel or perpendicular lines, you need to compare the two lines' slopes. This is done by comparing the two equations when they are both in the standard linear form: y = mx + b
...
Eqn 1
6x + 12y = 24 can be rewritten to be:
12y = -6x + 24
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divide both sides by 12:
y = -1/2 x + 2
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Eqn 2
x - y/2 = 18 can be rewritten to be:
-y/2 = -x + 18
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mulitiply both sides by -2:
y = 2x - 36
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Now we have both equations in standard form:
y = -1/2x + 2
y = 2x - 36
...
Referring back to the standard form, m, the coefficient of x, is the slope of the line. In analyzing whether two lines are parallel or perpendicular, we only need look at the slopes. We don't really care where either intersects the y axis (recall b = the y intercept), nor where they intersect each other.
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If m1 (slope of 1st equation) and m2 (slope of the 2nd equation) are equal, then the lines are parallel.
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We have:
m1 = -1/2
m2 = 2
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They are definitely not equal, so they are NOT parallel lines. The question only asks if they are parallel or perpendicular. They are not parallel, which would suggest they must be perpendicular, however, in the real world, lines can intersect without being perpendicular. So we should check.
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The test for two perpendicular lines is with their two slopes, one is the negative inverse of the other.
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Negative? One line will have positive slope, upward to the right at some angle, and its perpendicular complement cannot help but slope downward to the right at some angle.
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Inverse? For two intersecting lines, there are many possible angles between them, and thus many possible mathematical relationships between their slopes. However, only the inverse relationship will yield a 90 degree (perpendicular) angle.
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Going back to our two m's, m1 and m2:
m1 = -1/2 and m2 = 2; these are perpendicular slopes we just discussed.
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The answer to the question is the two lines are perpendicular.
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cheers,
Lee
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