What is the equation of the line that is perpendicular to and has the same $y$-intercept as the given line?
$y=\frac{1}{5} x+1$
Answer (1)
Determine the slope of the given line: 5 1 .
Calculate the slope of the perpendicular line: − 5 .
Identify the y-intercept of the given line: 1 .
Write the equation of the perpendicular line: y = − 5 x + 1 . The final answer is y = − 5 x + 1 .
Explanation
Understanding the Problem We are given the line y = 5 1 x + 1 and asked to find the equation of a line that is perpendicular to this line and has the same y -intercept.
Identifying Slope and Intercept The given line is in slope-intercept form, y = m x + b , where m is the slope and b is the y -intercept. For the given line, the slope is 5 1 and the y -intercept is 1.
Calculating Perpendicular Slope A line perpendicular to a line with slope m has a slope of − m 1 . Therefore, the slope of the perpendicular line is − 5 1 1 = − 5 .
Determining the y-intercept Since the perpendicular line has the same y -intercept as the given line, its y -intercept is also 1.
Writing the Equation Using the slope-intercept form y = m x + b , the equation of the perpendicular line is y = − 5 x + 1 .
Examples
Understanding perpendicular lines is crucial in various real-world applications, such as architecture and navigation. For instance, when designing a building, architects ensure that walls are perpendicular to the ground for stability. Similarly, in navigation, understanding perpendicular paths helps ships and airplanes maintain safe routes and avoid collisions. The concept of slope and y-intercept is also used in economics to model linear cost functions, where the slope represents the variable cost and the y-intercept represents the fixed cost.
