What is the slope of a line that is perpendicular to the line [tex]$2 y-3 x=8$[/tex]?
Answer (1)
Rewrite the given equation 2 y − 3 x = 8 in slope-intercept form to find its slope.
Identify the slope of the given line as 2 3 .
Calculate the negative reciprocal of the slope to find the slope of the perpendicular line.
The slope of the perpendicular line is − 3 2 .
Explanation
Understanding the Problem We are given the equation of a line 2 y − 3 x = 8 and asked to find the slope of a line perpendicular to it.
Finding the Slope of the Given Line First, we need to 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. To do this, we isolate y on one side of the equation:
2 y − 3 x = 8
Add 3 x to both sides:
2 y = 3 x + 8
Divide both sides by 2:
y = 2 3 x + 4
So, the slope of the given line is 2 3 .
Finding 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 our case, m = 2 3 , so the slope of the perpendicular line is:
− 2 3 1 = − 3 2
Final Answer Therefore, the slope of the line perpendicular to the given line is − 3 2 .
Examples
Understanding perpendicular slopes is crucial in various real-world applications, such as architecture and engineering. For instance, when designing a building, ensuring that walls are perpendicular to the ground is essential for structural stability. Similarly, in road construction, understanding perpendicular slopes helps in designing safe and efficient intersections. This concept also extends to navigation, where knowing the perpendicular direction to a path is vital for course correction and avoiding obstacles. By mastering the concept of perpendicular slopes, students gain a valuable tool for solving practical problems in diverse fields.
