What is the slope of a line that is perpendicular to [tex]$5 x-2 y=1$[/tex]?
Answer (2)
The slope of a line perpendicular to the line given by the equation 5 x − 2 y = 1 is − 5 2 . This was calculated by first rewriting the equation in slope-intercept form, finding its slope, and then determining the negative reciprocal. Thus, the answer is − 5 2 .
;
Rewrite the given equation 5 x − 2 y = 1 in slope-intercept form to find its slope.
The slope of the given line is 2 5 .
The slope of a line perpendicular to a line with slope m is − m 1 .
The slope of the perpendicular line is − 5 2 .
The final answer is − 5 2 .
Explanation
Understanding the Problem We are given the equation of a line 5 x − 2 y = 1 and we want to find the slope of a line that is perpendicular to it.
Finding the Slope of the Given Line First, we need to find the slope of the given line. To do this, we can rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Rewriting in Slope-Intercept Form Let's rewrite the equation 5 x − 2 y = 1 in slope-intercept form:
Subtract 5 x from both sides: − 2 y = − 5 x + 1
Divide both sides by − 2 : y = − 2 − 5 x + − 2 1
Simplify: y = 2 5 x − 2 1
So, the slope of the given line is 2 5 .
Finding 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 line perpendicular to a line with slope m is the negative reciprocal of m , which is − m 1 .
Calculating the Negative Reciprocal The slope of the given line is 2 5 , so the slope of a line perpendicular to it is:
− 2 5 1 = − 5 2
Final Answer Therefore, the slope of a line perpendicular to 5 x − 2 y = 1 is − 5 2 .
Examples
Understanding perpendicular slopes is crucial in various real-world applications, such as architecture and construction. For example, when designing a building, ensuring that walls are perpendicular to the ground is essential for structural stability. Similarly, in navigation, understanding perpendicular relationships helps in determining the shortest distance between two points or in calculating angles for precise maneuvering.
