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Difficulty: HardParallel and Perpendicular Lines

A straight path on a coordinate map is modeled by the equation 2x−y=82x - y = 8. A second path, which is perpendicular to the first path, passes through the point (3,5)(3, 5). What is the yy-intercept of the second path?

  1. A
    −1-1
  2. B
    72\frac{7}{2}
  3. 132\frac{13}{2}Answer
  4. D
    332\frac{33}{2}
  5. E
    83\frac{8}{3}

Answer

The y-intercept of the second path is 13/2
The y-intercept is 13/2. First, the slope of the given line is found to be 2. The perpendicular line must have a slope of -1/2. Substituting this slope and the point (3, 5) into the slope-intercept form gives the y-intercept of 13/2.

Step-by-Step Solution

1
Find the slope of the first path by rewriting the equation in slope-intercept form (y=mx+by = mx + b).
The equation 2x−y=82x - y = 8 becomes y=2x−8y = 2x - 8, indicating the slope m1=2m_1 = 2.
Knowing the slope of the first path allows us to determine the slope of any perpendicular path.
2
Calculate the slope of the perpendicular path by taking the negative reciprocal of the first path's slope.
The perpendicular slope m2=−12m_2 = -\frac{1}{2}.
Perpendicular lines have slopes that are negative reciprocals of one another.
3
Use the point-slope formula with the point (3,5)(3, 5) and the perpendicular slope −12-\frac{1}{2} to write the equation of the second path.
The equation is y−5=−12(x−3)y - 5 = -\frac{1}{2}(x - 3), which simplifies to y=−12x+32+5y = -\frac{1}{2}x + \frac{3}{2} + 5.
This establishes the linear relationship of the second path containing the point (3, 5).
4
Find the y-intercept by setting x=0x = 0 and simplifying.
Setting x=0x = 0 gives y=32+5=132y = \frac{3}{2} + 5 = \frac{13}{2}.
The y-intercept of a line is the y-coordinate where the line crosses the y-axis (when x = 0).

Key Concept

Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other, and their equations can be solved using point-slope form.
Estimated Time:1m 30s
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Topics

Parallel and Perpendicular Lines

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