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Math  /  Algebra

QuestionCity planners want to construct a bike path perpendicular to Aurora Avenue at point PP. An equation for the Aurora Avenue line is y=−27xy=-\frac{2}{7} x. Find an equation for the line for the bike path.
The equation of the line for the bike path is □\square . (Simplify your answer. Type an equation. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)

Studdy Solution

STEP 1

1. The line representing Aurora Avenue is given by the equation y=−27x y = -\frac{2}{7}x .
2. The bike path needs to be perpendicular to the Aurora Avenue line.
3. We need to find the equation of the line for the bike path in slope-intercept form.
4. The point P P is on the Aurora Avenue line.

STEP 2

1. Determine the slope of the bike path.
2. Use the point P P to find the y-intercept of the bike path.
3. Write the equation of the bike path in slope-intercept form.

STEP 3

Determine the slope of the bike path.
The slope of Aurora Avenue is m=−27 m = -\frac{2}{7} .
For two lines to be perpendicular, the product of their slopes must be −1-1. Therefore, the slope of the bike path mbike m_{\text{bike}} is the negative reciprocal of −27-\frac{2}{7}.
mbike=−1−27=72 m_{\text{bike}} = -\frac{1}{-\frac{2}{7}} = \frac{7}{2}

STEP 4

Use the point P P to find the y-intercept of the bike path.
Since the problem does not provide the exact coordinates of point P P , we will assume P P is a point on the line y=−27x y = -\frac{2}{7}x . Let's assume P P is at (x0,y0) (x_0, y_0) .
The equation of the bike path in point-slope form is:
y−y0=72(x−x0) y - y_0 = \frac{7}{2}(x - x_0)

STEP 5

Convert the equation to slope-intercept form.
Solve for y y :
y=72x−72x0+y0 y = \frac{7}{2}x - \frac{7}{2}x_0 + y_0
This equation is in slope-intercept form, where the slope is 72 \frac{7}{2} and the y-intercept is −72x0+y0 -\frac{7}{2}x_0 + y_0 .

STEP 6

Write the equation of the bike path in slope-intercept form.
Since we don't have specific coordinates for P P , the general equation for the bike path is:
y=72x+b y = \frac{7}{2}x + b
where b b is the y-intercept determined by the specific point P P .

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