Equations of parallel and perpendicular lines |
We checked with our algebra guru, and she told us that it's easy: "Parallel lines have the same slope but different X- and Y-intercepts. The slopes of perpendicular lines are negative reciprocals of each other." Grant, not surprisingly, managed to get the first half of that answer correct, although his route to a solution is rather hard to follow:
- "Write the equation for the first line and identify the slope and y-intercept.
- Example: y = 4x + 3 m = slope = 4 b = y-intercept = 3
- Copy the first half of the equation for the parallel line. A line is parallel to another if their slopes are identical.
- Example: Original line: y = 4x + 3 Parallel line: y = 4x
- Choose a y-intercept different from the original line."
"Regardless of the magnitude of the new y-intercept, as long as the slope is identical, the two lines will be parallel."While true, McKenzie's use of the word "magnitude" isn't quite correct: he really should have said, "value." That's just quibbling, though. It's when he gets to writing a perpendicular that Grant got in real trouble.
The real answer is that a perpendicular to a line of slope m has a slope of -1/m. It's that simple. Since his line's formula is y = 4x+3, the slope is 4. The negative reciprocal of 4 is -¼, so the equation must be y = -x/4 + 3, or 4y = -x + 12. These two lines will cross at the intercept; to get them to cross at any other point change the value of b from 12. McKenzie's instructions? Puzzle through this stuff:
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