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 4 x − 5 y = 5 by converting it to slope-intercept form: y = 5 4 x − 1 , so the slope is 5 4 .
Determine the slope of the perpendicular line by taking the negative reciprocal of the given line's slope: m ⊥ = − 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 the equation to standard form: 5 x + 4 y = 37 , so the final answer is 5 x + 4 y = 37 .
Explanation
Problem Analysis We are given a line and a point, and we want to find the equation of a line that is perpendicular to the given line and passes through the given point.
Finding 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 . We can rewrite this equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Slope-Intercept Form To rewrite the equation in slope-intercept form, we solve for y :
4 x − 5 y = 5 − 5 y = − 4 x + 5 y = − 5 − 4 x + − 5 5 y = 5 4 x − 1 So, the slope of the given line is 5 4 .
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. Therefore, the slope of the perpendicular line is: m ⊥ = − 5 4 1 = − 4 5
Point-Slope Form Now, we 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. We are given the point ( 5 , 3 ) , so x 1 = 5 and y 1 = 3 . The slope of the perpendicular line is − 4 5 . Plugging these values into the point-slope form, we get: y − 3 = − 4 5 ( x − 5 )
Standard Form Now, we convert the equation to standard form, which is A x + B y = C . First, we multiply both sides of the equation by 4 to eliminate the fraction: 4 ( y − 3 ) = 4 ( − 4 5 ( x − 5 ) ) 4 y − 12 = − 5 ( x − 5 ) 4 y − 12 = − 5 x + 25 Now, we rearrange the equation to get it in standard form: 5 x + 4 y = 25 + 12 5 x + 4 y = 37
Final Answer Therefore, 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 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For instance, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. If a wall deviates from perpendicularity, even by a small angle, it can compromise the structural integrity of the building. The equation of a line perpendicular to another line helps architects and engineers calculate and verify these angles, ensuring buildings are safe and stable. This principle extends to various other applications, such as designing bridges, roads, and even furniture, where perpendicularity plays a key role in functionality and aesthetics.
The equation of the line that is perpendicular to the line given by 4x - 5y = 5 and passes through the point (5, 3) is 5x + 4y = 37. This was found by determining the slope of the given line, finding the perpendicular slope, and using the point-slope form to derive the equation. The final result is given in standard form.
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