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Difficulty: EasyParallel and Perpendicular Lines

On a coordinate grid, line dd is represented by the equation y=−34x+5y = -\frac{3}{4}x + 5. Line ee is perpendicular to line dd and passes through the point (1,2)(1, 2). Which of the following is the equation of line ee?

  1. A
    y=−43x+103y = -\frac{4}{3}x + \frac{10}{3}
  2. B
    y=−34x+114y = -\frac{3}{4}x + \frac{11}{4}
  3. y=43x+23y = \frac{4}{3}x + \frac{2}{3}Answer
  4. D
    y=43x+1y = \frac{4}{3}x + 1
  5. E
    y=34x+54y = \frac{3}{4}x + \frac{5}{4}

Answer

The equation of line ee is y=43x+23y = \frac{4}{3}x + \frac{2}{3}.
The correct equation has a slope of 43\frac{4}{3} and a y-intercept of 23\frac{2}{3}. The slope of the given line is −34-\frac{3}{4}, meaning any line perpendicular to it must have a slope that is the negative reciprocal, which is 43\frac{4}{3}. Using the slope-intercept form y=mx+by = mx + b with the point (1,2)(1, 2) allows us to solve for bb by calculating 2=43(1)+b2 = \frac{4}{3}(1) + b, which simplifies to b=2−43=23b = 2 - \frac{4}{3} = \frac{2}{3}. Writing this together in slope-intercept form yields y=43x+23y = \frac{4}{3}x + \frac{2}{3}.

Step-by-Step Solution

1
Identify the slope of the given line dd.
The slope of line dd is −34-\frac{3}{4}.
The equation y=−34x+5y = -\frac{3}{4}x + 5 is in slope-intercept form (y=mx+by = mx + b), where the coefficient of xx represents the slope.
2
Determine the slope of the perpendicular line ee.
The slope of line ee is 43\frac{4}{3}.
Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of −34-\frac{3}{4} is 43\frac{4}{3}.
3
Substitute the perpendicular slope and the given point (1,2)(1, 2) into the slope-intercept equation to solve for the y-intercept bb.
b=23b = \frac{2}{3}
Plugging the values into y=mx+by = mx + b gives 2=43(1)+b2 = \frac{4}{3}(1) + b. Solving for bb requires subtracting 43\frac{4}{3} from 22, which yields 2−43=63−43=232 - \frac{4}{3} = \frac{6}{3} - \frac{4}{3} = \frac{2}{3}.
4
Write the final equation of line ee in slope-intercept form.
y=43x+23y = \frac{4}{3}x + \frac{2}{3}
Substituting the slope m=43m = \frac{4}{3} and y-intercept b=23b = \frac{2}{3} into the standard slope-intercept form equation.

Key Concept

Finding the equation of a line perpendicular to a given line through a given point using negative reciprocal slopes.
Estimated Time:1m 0s
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Topics

Parallel and Perpendicular Lines

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