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In Mathematics / College | 2025-07-03

What is the equation of the line that is perpendicular to and has the same $y$-intercept as the given line?

$y=\frac{1}{5} x+1$
$y=\frac{1}{5} x+5$
$y=5 x+1$
$y=5 x+5

Asked by donaldkimble857

Answer (2)

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 .
Construct the equation of the perpendicular line using the slope-intercept form: y = − 5 x + 1 . The final answer is y = − 5 x + 1 ​ .

Explanation

Understanding the Problem The problem asks us to find the equation of a line that is perpendicular to a given line and shares the same y-intercept. Let's break this down step by step.

Identifying Key Information The given line is y = 5 1 ​ x + 1 . We need to find a line perpendicular to this one that has the same y-intercept.

Finding the Perpendicular Slope The slope of the given line is 5 1 ​ . The slope of a line perpendicular to this will be the negative reciprocal of 5 1 ​ , which is − 5 .

Determining the Y-Intercept The y-intercept of the given line is 1 . The new line must have the same y-intercept, so its y-intercept is also 1 .

Constructing the Equation Now we use the slope-intercept form of a line, which is y = m x + b , where m is the slope and b is the y-intercept. We have m = − 5 and b = 1 . Substituting these values, we get y = − 5 x + 1 .

Final Answer Therefore, the equation of the line that is perpendicular to y = 5 1 ​ x + 1 and has the same y-intercept is y = − 5 x + 1 .


Examples
Understanding perpendicular lines is crucial in many real-world applications. For example, architects use this concept to ensure walls are perfectly upright, and streets intersect at right angles for efficient traffic flow. Imagine you're designing a garden and want a path that's perpendicular to an existing fence. Knowing the slope of the fence allows you to calculate the slope of the path, ensuring they meet at a 90-degree angle. This principle also applies in navigation, where understanding perpendicular bearings helps ships and airplanes stay on course.

Answered by GinnyAnswer | 2025-07-03

The equation of the line that is perpendicular to the given line y = 5 1 ​ x + 1 , while having the same y-intercept, is y = − 5 x + 1 .
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Answered by Anonymous | 2025-07-04

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