Identify an equation in point-slope form for the line perpendicular to $y=3 x+5$ that passes through $(4,-1)$.
A. $y+1=3(x-4)$
B. $y-4=-\frac{1}{3}(x+1)$
C. $y+1=-\frac{1}{3}(x-4)$
D. $y-1=-3(x+4)$
Answer (1)
Determine the slope of the given line: m = 3 .
Calculate the slope of the perpendicular line: m ⊥ = − 3 1 .
Use the point-slope form with the point ( 4 , − 1 ) and the perpendicular slope: y + 1 = − 3 1 ( x − 4 ) .
Identify the correct equation from the options: y + 1 = − 3 1 ( x − 4 )
Explanation
Find the slope of the given line First, we need to find the slope of the given line, which is in slope-intercept form ( y = m x + b ). The given line is y = 3 x + 5 , so its slope is 3 .
Determine the slope of the perpendicular line Next, we need to find the slope of the line perpendicular to the given line. The slope of a perpendicular line is the negative reciprocal of the original line's slope. So, the slope of the perpendicular line is − 3 1 .
Write the equation of the perpendicular line in point-slope form Now, we write the equation of the perpendicular line in point-slope form using the point ( 4 , − 1 ) and the slope − 3 1 . The point-slope form is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope. Substituting the values, we get y − ( − 1 ) = − 3 1 ( x − 4 ) , which simplifies to y + 1 = − 3 1 ( x − 4 ) .
Identify the correct option Finally, we compare the equation we obtained with the given options. The equation y + 1 = − 3 1 ( x − 4 ) matches option C.
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. The concept of point-slope form helps architects define the precise location and orientation of these lines, ensuring structural integrity and aesthetic appeal. Imagine designing a roof where the slope needs to be perpendicular to a supporting beam; using these principles ensures the roof is correctly aligned and stable.
