The equation of a line is [tex]$y=-5 x+2$[/tex]. What is the slope of a line that is perpendicular to this line?
Answer (1)
Identify the slope of the given line: The slope of y = − 5 x + 2 is − 5 .
Calculate the negative reciprocal: The negative reciprocal of − 5 is − − 5 1 .
Simplify the negative reciprocal: − − 5 1 = 5 1 .
State the final answer: The slope of the perpendicular line is 5 1 .
Explanation
Understanding the Problem We are given the equation of a line: y = − 5 x + 2 . Our goal is to find the slope of a line that is perpendicular to this line.
Identifying the Slope of the Given Line The given equation is in slope-intercept form, y = m x + b , where m represents the slope and b represents the y-intercept. In our case, m = − 5 and b = 2 . So, the slope of the given line is − 5 .
Understanding Perpendicular Slopes The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. If the slope of the given line is m , then the slope of the perpendicular line is − m 1 .
Calculating the Perpendicular Slope In our case, the slope of the given line is m = − 5 . Therefore, the slope of the perpendicular line is − − 5 1 = 5 1 .
Examples
Understanding perpendicular slopes is crucial in various real-world applications. For instance, architects use this concept to design buildings with walls that are perfectly perpendicular to the ground, ensuring structural stability. Similarly, in navigation, knowing the perpendicular direction to a path helps in determining the shortest route to a destination. In computer graphics, perpendicular vectors are used to calculate lighting and shading effects, creating realistic images. This mathematical principle ensures precision and accuracy in many practical scenarios.
