What is the equation of the line that is perpendicular to the line [tex]y=\frac{3}{5} x+10[/tex] and passes through the point (15,-5)?
A. [tex]y=\frac{3}{5} x-20[/tex]
B. [tex]y=-\frac{3}{5} x+20[/tex]
C. [tex]y=\frac{5}{3} x-20[/tex]
D. [tex]y=-\frac{5}{3} x+20[/tex]
Answer (2)
Determine the slope of the given line: m 1 = 5 3 .
Calculate the slope of the perpendicular line: m 2 = − 3 5 .
Use the point-slope form with the point ( 15 , − 5 ) : y − ( − 5 ) = − 3 5 ( x − 15 ) .
Convert to slope-intercept form: y = − 3 5 x + 20 .
Explanation
Problem Analysis We are given the equation of a line y = 5 3 x + 10 and a point ( 15 , − 5 ) . We need to find the equation of the line that is perpendicular to the given line and passes through the given point.
Finding the Slope of the Given Line The given line is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. The slope of the given line is 5 3 .
Finding the Slope of the Perpendicular Line The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. Therefore, the slope of the perpendicular line is − 5 3 1 = − 3 5 .
Using Point-Slope Form Now we use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. We have the point ( 15 , − 5 ) and the slope − 3 5 . Plugging these values into the point-slope form, we get:
y − ( − 5 ) = − 3 5 ( x − 15 )
Simplifying to Slope-Intercept Form Now, we simplify the equation and rewrite it in slope-intercept form, y = m x + b :
y + 5 = − 3 5 x + 3 5 ( 15 )
y + 5 = − 3 5 x + 25
y = − 3 5 x + 25 − 5
y = − 3 5 x + 20
Final Answer The equation of the line that is perpendicular to the line y = 5 3 x + 10 and passes through the point ( 15 , − 5 ) is y = − 3 5 x + 20 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, architects need to ensure that walls are perpendicular to the ground for stability. If a wall deviates from being perfectly perpendicular, it can compromise the structural integrity of the building. The equation of a line perpendicular to another line helps in accurately planning and executing such designs, ensuring safety and longevity of the structure. In this case, if the ground has a slope represented by y = 5 3 x + 10 , the supporting wall must follow the equation y = − 3 5 x + 20 to be perfectly perpendicular.
The equation of the line that is perpendicular to y = 5 3 x + 10 and passes through the point ( 15 , − 5 ) is y = − 3 5 x + 20 . This line has a slope of − 3 5 , which is the negative reciprocal of the original slope 5 3 . The corresponding answer choice is D.
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