Which equations represent the line that is perpendicular to the line [tex]$5 x-2 y=-6$[/tex] and passes through the point [tex]$(5,-4)$[/tex]? Select three options.
[tex]$y=\frac{2}{5} x-2$[/tex]
[tex]$2 x+5 y=-10$[/tex]
[tex]$2 x-5 y=-10$[/tex]
[tex]$y+4=-\frac{2}{5}(x-5)$[/tex]
[tex]$y-4=\frac{5}{2}(x+5)$[/tex]
Answer (1)
Find the slope of the given line 5 x − 2 y = − 6 by converting it to slope-intercept form: y = 2 5 x + 3 , so the slope is 2 5 .
Determine the slope of the perpendicular line by taking the negative reciprocal of the original slope: m ⊥ = − 5 2 .
Use the point-slope form of a line with the perpendicular slope and the given point ( 5 , − 4 ) : y + 4 = − 5 2 ( x − 5 ) .
Convert to standard form: 2 x + 5 y = − 10 .
y + 4 = − 5 2 ( x − 5 ) , 2 x + 5 y = − 10
Explanation
Find the slope of the given line First, let's find the slope of the given line 5 x − 2 y = − 6 . To do this, we'll rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Rewrite in slope-intercept form We start with 5 x − 2 y = − 6 . Subtract 5 x from both sides to get − 2 y = − 5 x − 6 . Then, divide both sides by − 2 to solve for y : y = − 2 − 5 x − 6 = 2 5 x + 3 So, the slope of the given line is 2 5 .
Determine the slope of the perpendicular line Now, we need to find the slope of a 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 ⊥ = − 2 5 1 = − 5 2
Use the point-slope form Next, we'll use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the point the line passes through. We are given the point ( 5 , − 4 ) , so x 1 = 5 and y 1 = − 4 . Plugging in the slope m = − 5 2 and the point ( 5 , − 4 ) , we get: y − ( − 4 ) = − 5 2 ( x − 5 ) y + 4 = − 5 2 ( x − 5 ) This matches one of the given options.
Convert to slope-intercept form Let's convert the point-slope form to slope-intercept form to see if we can find another matching option: y + 4 = − 5 2 x + 5 2 ( 5 ) y + 4 = − 5 2 x + 2 y = − 5 2 x + 2 − 4 y = − 5 2 x − 2
Convert to standard form Now, let's convert the slope-intercept form to standard form A x + B y = C : y = − 5 2 x − 2 Multiply both sides by 5: 5 y = − 2 x − 10 Add 2 x to both sides: 2 x + 5 y = − 10 This matches another one of the given options.
Final Answer The three equations that represent the line perpendicular to 5 x − 2 y = − 6 and passing through the point ( 5 , − 4 ) are:
y + 4 = − 5 2 ( x − 5 )
2 x + 5 y = − 10
y = − 5 2 x − 2 (This option was not given, but it is equivalent to the other two)
Select the correct options The equations that represent the line perpendicular to the line 5 x − 2 y = − 6 and passes through the point ( 5 , − 4 ) are:
y + 4 = − 5 2 ( x − 5 )
2 x + 5 y = − 10
Examples
Understanding perpendicular lines is crucial in various real-world applications, such as architecture and navigation. For example, architects use perpendicular lines to ensure that walls are at right angles, providing structural stability to buildings. Similarly, in navigation, understanding perpendicular relationships helps in plotting courses and determining the shortest distance between two points. This problem demonstrates how to find the equation of a line perpendicular to another, which is a fundamental concept in these fields.
For instance, imagine you're designing a rectangular garden where one side is defined by the line 5 x − 2 y = − 6 . To ensure the adjacent side is perfectly perpendicular and passes through a specific point ( 5 , − 4 ) , you would use the methods described above to find the equation of the perpendicular line. This ensures your garden has the precise right angles needed for its design.
