SOLUTION: Determine whether each pair of lines is parallel, perpendicular, or neither. (a) 3x + 4y = 12; 4x - 3y = -12 (b) y = 1; y = -1

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Question 1207841: Determine whether each pair of lines is parallel, perpendicular, or neither.
(a) 3x + 4y = 12; 4x - 3y = -12

(b) y = 1; y = -1
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
.
Determine whether each pair of lines is parallel, perpendicular, or neither.
(a) 3x + 4y = 12; 4x - 3y = -12
(b) y = 1; y = -1
~~~~~~~~~~~~~~~~~~~~~~~~ 

(a)  Line 3x + 4y = 12  is the same as  y = .

     From this form, you see that the slope of this line is  = .


     Line 4x - 3y = -12  is the same as  y = .

     From this form, you see that the slope of this line is  = .


     The slopes    and   are reciprocal: their product is -1.

     It means that the lines are perpendicular.    ANSWER




(b)  Line y = 1 is horizontal line parallel to x-axis.

     Line y = -1 is another/different horizontal line parallel to x-axis.

     Hence, lines y = 1  and  y = -1 are parallel.    ANSWER


      They both have the same slope of 0.

Solved.



Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


For the first one, the other tutor uses a basic algebraic method, showing the lines are perpendicular by finding the slopes and showing that the product of the slopes is -1.

A more sophisticated (and faster and easier) method is to use a dot product.

Given two linear equations ax+by=m and cx+dy=n, the dot product is ac+bd -- the product of the x coefficients plus the product of the y coefficients. The lines are perpendicular only if the dot product is 0.

In the first example, the dot product is (3)(4)+(4)(-3) = 12-12 = 0, so the lines are perpendicular.

With a little experience, you can tell that the lines are perpendicular by inspection. By comparing the two equations, we see that the coefficients of x and y are switched, with one of them changing sign. That guarantees that the dot product will be 0 and the lines will be perpendicular.

Here are a couple of quick examples of pair of equations of lines that are perpendicular:
3x-7y=4 and 7x+3y=4
17x+39y=100 and 39x-17y=0
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