Discover The Equation Of A Line Perpendicular To The Green Sine And With X-intercept 5

In mathematics, particularly in coordinate geometry, determining the equation of a line that satisfies specific conditions is a fundamental skill. This article delves into a common problem: finding the equation of a line that is perpendicular to a given line (in this case, a 'green sine,' which we will interpret as a line with a specific slope) and possesses a specific x-intercept. This type of problem integrates concepts such as slopes of perpendicular lines, the slope-intercept form of a linear equation, and the significance of intercepts.

Understanding the Problem: Perpendicularity and Intercepts

Before diving into the solution, it's crucial to grasp the underlying principles. Two lines are perpendicular if they intersect at a right angle (90 degrees). The slopes of perpendicular lines have a unique relationship: they are negative reciprocals of each other. That is, if one line has a slope m, a line perpendicular to it will have a slope of -1/m. This property is the cornerstone of solving this problem.

The x-intercept is the point where a line crosses the x-axis. At this point, the y-coordinate is always zero. Knowing the x-intercept provides us with a specific point (x, 0) that the line passes through, which is essential for determining the equation of the line.

The Slope-Intercept Form: A Key Tool

The slope-intercept form of a linear equation, y = mx + b, is the primary tool we'll use. Here, m represents the slope of the line, and b represents the y-intercept (the point where the line crosses the y-axis). By determining the slope and a point on the line, we can substitute these values into the slope-intercept form and solve for the y-intercept, thus defining the complete equation of the line. To effectively tackle this problem, consider the significance of the perpendicular slope. The slope of the line perpendicular to the green sine wave is the negative reciprocal of the green sine wave's slope. Understanding this relationship is pivotal in correctly determining the new line's slope. Next, the x-intercept is the point where the line crosses the x-axis. This is a crucial piece of information because it provides a specific coordinate point (x, 0) that the line passes through. This point will be used to solve for the y-intercept in the line's equation.

Finally, the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept, is the framework within which we'll construct our equation. By finding the slope (using the perpendicularity condition) and a point on the line (the x-intercept), we can substitute these values into the slope-intercept form and solve for the y-intercept.

Decoding the "Green Sine" and the Given X-Intercept

The problem states that the line we are looking for is perpendicular to the "green sine." While "green sine" might sound ambiguous, in this mathematical context, it is likely referring to a line with a slope. Since the options provided have slopes expressed as fractions, let's assume the "green sine" has a slope that is the negative reciprocal of the slopes in the answer choices. This means we need to figure out what slope would result in the given answer choices when we take its negative reciprocal.

We also know that the line has an x-intercept of 5. This means the line passes through the point (5, 0). This information is crucial because it gives us a specific point on the line, which we can use to solve for the y-intercept once we've determined the slope. The phrase "equation of the line" immediately directs our focus to the algebraic representation of a straight line. This typically takes the form of the slope-intercept equation, y = mx + b, or the point-slope equation, y - y1 = m(x - x1). In this case, given that we're dealing with perpendicularity and an x-intercept, the slope-intercept form will likely be the most convenient to use.

The Significance of the Slope

The slope, denoted by m in the equation y = mx + b, dictates the steepness and direction of the line. A positive slope indicates that the line rises as you move from left to right, while a negative slope indicates that the line falls. The magnitude of the slope also matters; a larger absolute value indicates a steeper line. This slope plays a critical role when determining perpendicularity. The slopes of perpendicular lines are negative reciprocals of each other. This means that if we can determine the slope of the "green sine," we can easily find the slope of the line perpendicular to it.

The Role of the X-Intercept

The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. The given x-intercept of 5 provides us with the point (5, 0) on the line. This point is invaluable because it allows us to determine the y-intercept (b) in the slope-intercept form once we know the slope. By substituting the x and y coordinates of the x-intercept, along with the slope, into the equation y = mx + b, we can solve for b. This exemplifies the practical use of the x-intercept in defining the line's characteristics and ultimately deriving its equation.

Solving for the Equation: A Step-by-Step Approach

Now, let's systematically solve the problem. We have four options for the equation of the line:

A. y = -3/4 x + 8 B. y = -3/4 x + 6 C. y = 4/3 x - 8 D. y = 4/3 x - 6

First, we need to identify the slope of the line in each option. The slope is the coefficient of x in the slope-intercept form. Let's analyze each option:

A. Slope = -3/4 B. Slope = -3/4 C. Slope = 4/3 D. Slope = 4/3

Since the line we are looking for is perpendicular to the "green sine," we need to find the negative reciprocal of the slope of the "green sine." Let's assume, for now, that the "green sine" has a slope of -3/4. The negative reciprocal of -3/4 is 4/3. This means options C and D are potential candidates. If we assume the