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Answers:

  1. Parallel slope =  2; Perpendicular slope = -1/2
  2. Parallel slope =  1/3; Perpendicular slope = -3
  3. Parallel slope =  4/3; Perpendicular slope = -3/4
  4. Parallel slope =  -2; Perpendicular slope = 1/2
  5. Parallel slope =  2/3; Perpendicular slope = -3/2
  6. Parallel slope =  -4; Perpendicular slope = 1/4
  7. Parallel slope =  4; Perpendicular slope = -1/4
  8. Parallel slope =  3; Perpendicular slope = -1/3
  9. Parallel slope =  1/4; Perpendicular slope = -4
  10. Parallel slope =  1/2; Perpendicular slope = -2
  11. Parallel slope =  -3; Perpendicular slope = 1/3
  12. Parallel slope =  3/4; Perpendicular slope = -4/3
  13. Parallel slope =  -1/3; Perpendicular slope = 3
  14. Parallel slope = -1/2; Perpendicular slope = 2
  15. Parallel slope = -1/4; Perpendicular slope = 4
  16. Parallel slope = -3/4; Perpendicular slope = 4/3
  17. Parallel slope = -3/2; Perpendicular slope = 2/3
  18. Parallel slope = -4/3; Perpendicular slope = 3/4

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Explanation:

Parallel lines always have equal slopes, but different y intercepts. For example, the two lines y = 3x+5 and y = 3x+7 have a slope of 3. Refer to y = mx+b where m is the slope. Parallel lines never intersect.

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For perpendicular lines, the two slopes always multiply to -1 assuming neither line is vertical nor horizontal. Consider the two linear equations

y = (1/2)x+5

y = -2x-7

The slopes 1/2 and -2 multiply to -1 which is enough to show that they are perpendicular lines. The two lines meet at a 90 degree angle.

To go from one slope to the other, we apply the negative reciprocal. That means we...

  • flip the fraction
  • flip the sign

Applying the first step to 1/2 gets us 2/1 or 2. Then changing the sign has us go from 2 to -2. This process can be reversed.

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For some of these equations, you'll need to rearrange things into either point-slope form or slope intercept form. This will allow you to read off the slope.

Point slope form is y-y1 = m(x-x1)

Slope intercept form is y = mx+b

For either case, all we care about is the variable m.

I'll show how problem 14 transforms into slope intercept form

x+2y = 14

2y = -x+14

y = (-x+14)/2

y = (-x/2) + (14/2)

y = (-1/2)x + 7

We see that m = -1/2 here. Anything parallel to this will have the same slope. The perpendicular slope is 2 after applying the negative reciprocal operation. Problems 13, 15, 16, 17, and 18 are going to follow similar steps.

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