What is the equation of the line that is perpendicular to the given line and passes through the point $(5,3)$?

$4 x-5 y=5$
$5 x+4 y=37$
$4 x+5 y=5$
$5 x-4 y=8$

Find the slope of the given line by converting it to slope-intercept form: y = 5 4 ​ x − 1 , so the slope m 1 ​ = 5 4 ​ .
Determine the slope of the perpendicular line: m 2 ​ = − m 1 ​ 1 ​ = − 4 5 ​ .
Use the point-slope form of a line with the point ( 5 , 3 ) and the perpendicular slope: y − 3 = − 4 5 ​ ( x − 5 ) .
Convert to standard form: 5 x + 4 y = 37 ​ .

Explanation

Problem Analysis We are given the equation of a line 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. The equation of the given line is 4 x − 5 y = 5 . To find the slope, we can 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 can 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 m 1 ​ = 5 4 ​ .

Find the slope of the perpendicular line Next, we need to find the slope of the line that is perpendicular to the given line. The slope of a line perpendicular to a line with slope m 1 ​ is the negative reciprocal of m 1 ​ , which is m 2 ​ = − m 1 ​ 1 ​ .

In this case, m 1 ​ = 5 4 ​ , so the slope of the perpendicular line is:
m 2 ​ = − 5 4 ​ 1 ​ = − 4 5 ​

Use the point-slope form Now that we have the slope of the perpendicular line, m 2 ​ = − 4 5 ​ , and a point that the line passes through, ( 5 , 3 ) , we can use the point-slope form of a line to find the equation of the perpendicular line. 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.

Plugging in the values, we get:
y − 3 = − 4 5 ​ ( x − 5 )

Rewrite in standard form Finally, we can rewrite the equation in standard form, which is A x + B y = C . Starting with y − 3 = − 4 5 ​ ( x − 5 ) , we can multiply both sides by 4 to eliminate the fraction:

4 ( y − 3 ) = − 5 ( x − 5 )
4 y − 12 = − 5 x + 25
Now, we can rearrange the equation to get it in standard form:
5 x + 4 y = 25 + 12
5 x + 4 y = 37
So, the equation of the line that is perpendicular to the given line and passes through the point ( 5 , 3 ) is 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 ground is essential for stability. If a wall deviates from perpendicularity, it can compromise the structural integrity of the building. The equation of a line, especially in relation to perpendicularity, helps architects and engineers calculate and maintain the correct angles and alignments, ensuring the building is safe and sound. Imagine designing a roof where the rafters need to be perpendicular to a supporting beam; accurately calculating the slope and equation of these lines is vital for a stable roof structure.

Answered by GinnyAnswer | 2025-07-03

The equation of the line perpendicular to 4 x − 5 y = 5 that passes through ( 5 , 3 ) is 5 x + 4 y = 37 .
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Answered by Anonymous | 2025-07-04