Which equation represents a line which is perpendicular to the line [tex]$3 x-8 y=-16$[/tex]?
A. [tex]$y=\frac{8}{3} x+8$[/tex]
B. [tex]$y=-\frac{3}{8} x+1$[/tex]
C. [tex]$y=\frac{3}{8} x-4$[/tex]
D. [tex]$y=-\frac{8}{3} x-2$[/tex]
Answer (1)
Rewrite the given equation 3 x − 8 y = − 16 in slope-intercept form to find its slope.
Determine the slope of the perpendicular line by taking the negative reciprocal of the original slope.
Identify the answer choice with the slope that matches the negative reciprocal.
The equation of the perpendicular line is y = − 3 8 x − 2 .
Explanation
Find the slope of the given line. We are given the equation of a line 3 x − 8 y = − 16 and asked to find the equation of a line perpendicular to it. To do this, we first need to find the slope of the given line.
Rewrite in slope-intercept form. To find the slope, we rewrite the given equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Starting with 3 x − 8 y = − 16 , we isolate y :
Subtract 3 x from both sides: − 8 y = − 3 x − 16 Divide both sides by − 8 : y = − 8 − 3 x + − 8 − 16 Simplify: y = 8 3 x + 2
So, the slope of the given line is 8 3 .
Find the slope of the perpendicular line. The slope of a line perpendicular to a line with slope m is the negative reciprocal of m , which is − m 1 . In this case, the slope of the perpendicular line is:
− 8 3 1 = − 3 8
Identify the correct equation. Now we need to find which of the answer choices has a slope of − 3 8 . The answer choices are:
A) y = 3 8 x + 8 (slope is 3 8 )
B) y = − 8 3 x + 1 (slope is − 8 3 )
C) y = 8 3 x − 4 (slope is 8 3 )
D) y = − 3 8 x − 2 (slope is − 3 8 )
Only option D has the correct slope.
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's slope is represented by the equation y = 8 3 x + 2 , a supporting beam must be perpendicular, following the slope y = − 3 8 x − 2 to ensure structural integrity. This principle applies to various aspects of construction, from laying foundations to framing roofs, ensuring buildings are safe and sound.
