Find the equation of the line perpendicular to [tex]y=\frac{1}{3} x+7[/tex] that includes the point (2,4).
Give your answer in Point-Slope Form.
[tex]y-[?]=\square(x-\square)[/tex]
Point-Slope Form: [tex]y-y_1=m\left(x_1-x_1\right)[/tex]
Answer (2)
Determine the slope of the given line: The slope of y = 3 1 x + 7 is 3 1 .
Calculate the slope of the perpendicular line: The negative reciprocal of 3 1 is − 3 .
Apply the point-slope form: Using the point ( 2 , 4 ) and the slope − 3 , the equation is y − 4 = − 3 ( x − 2 ) .
State the final equation in point-slope form: y − 4 = − 3 ( x − 2 )
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is perpendicular to a given line and passes through a specific point. We need to express the equation in point-slope form. Let's break this down step by step.
Finding the Slope of the Given Line The given line is y = 3 1 x + 7 . The slope of this line is 3 1 .
Finding the Slope of the Perpendicular Line The slope of a line perpendicular to the given line is the negative reciprocal of the given line's slope. So, the slope of the perpendicular line is m = − 3 1 1 = − 3 .
Using the Point-Slope Form We are given the point ( 2 , 4 ) that the perpendicular line passes through. Now we can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope.
Substituting Values into Point-Slope Form Substituting the point ( 2 , 4 ) and the slope m = − 3 into the point-slope form, we get: y − 4 = − 3 ( x − 2 ) .
Final Answer Therefore, the equation of the line perpendicular to y = 3 1 x + 7 that includes the point ( 2 , 4 ) in point-slope form is y − 4 = − 3 ( x − 2 ) .
Examples
Imagine you're designing a ramp that needs to be perpendicular to a path. The path's slope is like the slope of the given line, and the point (2,4) is where the ramp needs to connect. Finding the equation of the perpendicular line helps you determine the correct angle and position for the ramp to ensure it meets the path perfectly. This is a practical application of understanding perpendicular lines and their equations.
The equation of the line that is perpendicular to y = 3 1 x + 7 and passes through the point ( 2 , 4 ) is y − 4 = − 3 ( x − 2 ) in point-slope form. The slope of the given line is 3 1 , making the slope of the perpendicular line − 3 .
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