What is the equation of the line that is perpendicular to the given line and passes through the point $(5,3)$?
$4 x-5 y=5$
$5 x+4 y=37$
$4 x+5 y=5$
$5 x-4 y=8$
Answer (2)
Find the slope of the given line by converting it to slope-intercept form: y = 5 4 x − 1 , so the slope is 5 4 .
Calculate the slope of the perpendicular line by taking the negative reciprocal of the original slope: − 4 5 .
Use the point-slope form of a line with the point ( 5 , 3 ) and the perpendicular slope to find the equation: y − 3 = − 4 5 ( x − 5 ) .
Convert the equation to standard form: 5 x + 4 y = 37 .
Explanation
Problem Analysis We are given the equation of a line 4 x − 5 y = 5 and a point ( 5 , 3 ) . Our goal is to find the equation of the line that is perpendicular to the given line and passes through the given point.
Find the slope of the given line First, we need to find the slope of the given line. To do this, we rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Starting with 4 x − 5 y = 5 , we isolate y :
− 5 y = − 4 x + 5
y = − 5 − 4 x + 5
y = 5 4 x − 1
So, the slope of the given line is 5 4 .
Find the slope of the perpendicular line Next, we find the slope of the line perpendicular to the given line. The slope of a perpendicular line is the negative reciprocal of the original slope. If the original slope is m , the perpendicular slope is − m 1 .
In this case, the slope of the given line is 5 4 , so the slope of the perpendicular line is:
m ⊥ = − 5 4 1 = − 4 5
Use the point-slope form Now, we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. We are given the point ( 5 , 3 ) and the slope − 4 5 .
Plugging in the values, we get:
y − 3 = − 4 5 ( x − 5 )
Rewrite the equation in standard form We simplify the equation to get it into standard form, A x + B y = C :
y − 3 = − 4 5 x + 4 25
y = − 4 5 x + 4 25 + 3
y = − 4 5 x + 4 25 + 4 12
y = − 4 5 x + 4 37
To get rid of the fractions, we multiply the entire equation by 4:
4 y = − 5 x + 37
Finally, we rearrange the equation to get it into standard form:
5 x + 4 y = 37
Final Answer The equation of the line that is perpendicular to the given line 4 x − 5 y = 5 and passes through the point ( 5 , 3 ) is 5 x + 4 y = 37 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, ensuring that walls are perpendicular to the floor is essential for structural integrity. If a floor has a slope represented by the equation 4 x − 5 y = 5 , architects need to calculate the equation of a line perpendicular to it to ensure walls are built correctly. Using the point ( 5 , 3 ) as a reference, they can determine the wall's alignment using the equation 5 x + 4 y = 37 , ensuring the building's stability and safety.
The equation of the line perpendicular to 4 x − 5 y = 5 that passes through the point ( 5 , 3 ) is 5 x + 4 y = 37 .
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