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: y = 2 3 x + 4 .
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: − 2 3 1 = − 3 2 .
The slope of the line perpendicular to the given 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.
Converting to Slope-Intercept Form 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 in the equation 2 y − 3 x = 8 .
Isolating y Add 3 x to both sides of the equation: 2 y = 3 x + 8
Finding the Slope Divide both sides by 2: y = 2 3 x + 2 8 y = 2 3 x + 4
Determining the Perpendicular Slope From the slope-intercept form, we can see that the slope of the given line is 2 3 .
Calculating the Negative Reciprocal 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: − 2 3 1 = − 3 2
Final Answer Therefore, the slope of a line perpendicular to the line 2 y − 3 x = 8 is − 3 2 .
Examples
Understanding perpendicular slopes is crucial in various real-world applications, such as designing roads and bridges. Civil engineers use this concept to ensure that structures are built at the correct angles for stability and safety. For example, when designing a bridge, the support beams need to be perpendicular to the road surface to evenly distribute the load. By calculating the slopes and ensuring they are negative reciprocals of each other, engineers can guarantee the structural integrity of the bridge.
