Equation Of A Line With Slope 0 Passing Through (-2, -5)

In the realm of coordinate geometry, understanding the equations of lines is fundamental. This article delves into a specific scenario: determining the equation of a line that has a slope of 0 and passes through the point (-2, -5). This seemingly simple problem unveils key concepts about linear equations, slopes, and the graphical representation of lines. We will explore the different options presented and methodically arrive at the correct solution, reinforcing your understanding of linear equations and their properties.

Decoding the Slope-Intercept Form and the Significance of Slope 0

To tackle this problem effectively, let's first revisit the slope-intercept form of a linear equation: y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). The slope, m, quantifies the steepness of the line. A positive slope indicates an upward slant, a negative slope indicates a downward slant, a slope of 0 signifies a horizontal line, and an undefined slope represents a vertical line.

Now, let’s focus on the case at hand: a slope of 0. When m = 0, the equation y = mx + b simplifies to y = b. This is a crucial observation. It tells us that any line with a slope of 0 will have an equation of the form y = b, where b is a constant. This constant b is precisely the y-coordinate of every point on the line. This is because, with a slope of 0, the line neither rises nor falls; it runs horizontally, maintaining a constant y-value. Therefore, understanding that a slope of 0 implies a horizontal line with an equation y = b is the cornerstone to solving this problem.

The y-intercept, b, holds significant importance. It defines where the line intersects the y-axis. In the context of a horizontal line (slope 0), the y-intercept dictates the constant y-value that the line maintains across all x-values. For instance, if b = 3, the equation becomes y = 3, signifying a horizontal line passing through the point (0, 3) on the y-axis. Every point on this line will have a y-coordinate of 3, regardless of its x-coordinate. This fundamental characteristic of horizontal lines – a constant y-value – is key to accurately identifying their equations. In essence, the equation y = b encapsulates the essence of a horizontal line, where the slope is zero, and the y-value remains constant, determined by the y-intercept b. Visualizing this on a coordinate plane makes the concept even clearer. Imagine a line stretching infinitely to the left and right, perfectly parallel to the x-axis. This is the graphical representation of a line with a slope of 0. The y-intercept, b, simply shifts this line up or down the y-axis, but its horizontal orientation remains unchanged. This visual understanding reinforces the connection between the equation y = b and the geometric representation of a horizontal line.

Applying the Point (-2, -5) to Determine the Correct Equation

We know that the line passes through the point (-2, -5). This ordered pair provides us with the x and y coordinates of a specific location on the line. Remember, since the line has a slope of 0, its equation will be in the form y = b. The y-coordinate of the point (-2, -5) is -5. This directly corresponds to the value of b in our equation. Therefore, the equation of the line must be y = -5. This means that regardless of the x-coordinate, the y-coordinate of any point on this line will always be -5.

To solidify this concept, let’s consider a few additional points that would lie on this line. For example, the points (0, -5), (1, -5), and (-3, -5) all satisfy the equation y = -5. Notice how the y-coordinate remains constant at -5, while the x-coordinate varies. This is the defining characteristic of a horizontal line. Conversely, points such as (-2, 0), (-2, 1), or (-2, -4) would not lie on this line, as their y-coordinates are not equal to -5. This process of verifying points reinforces the understanding of how a linear equation defines the relationship between x and y coordinates on a line. The specific equation y = -5 dictates that only points with a y-coordinate of -5 can reside on this horizontal line. This connection between points, equations, and graphical representation is a cornerstone of coordinate geometry.

Visualizing the point (-2, -5) on a coordinate plane further clarifies the solution. Imagine plotting this point. Now, envision a horizontal line passing through this point. This line stretches infinitely to the left and right, maintaining a constant y-value of -5. This visual representation directly corresponds to the equation y = -5. The line runs parallel to the x-axis, intersecting the y-axis at the point (0, -5). This graphical interpretation solidifies the understanding that the equation y = -5 accurately represents a horizontal line passing through the given point.

Analyzing the Incorrect Options

Let's examine why the other options are incorrect, further solidifying our understanding of linear equations and slopes.

• Option B: x = -5. This equation represents a vertical line, not a horizontal line. Vertical lines have an undefined slope, not a slope of 0. The equation x = -5 indicates that every point on the line has an x-coordinate of -5, regardless of the y-coordinate. This line would pass through the point (-5, 0) on the x-axis and run parallel to the y-axis. Therefore, it does not fit the criteria of a line with a slope of 0.

• Option C: y = -2. This equation represents a horizontal line, but it passes through the point (0, -2), not (-2, -5). While it has a slope of 0, it does not satisfy the condition of passing through the given point. The equation y = -2 indicates that every point on this line has a y-coordinate of -2. It would run parallel to the x-axis but would be located at a different vertical position than the line we are seeking.

• Option D: x = -2. Similar to option B, this equation represents a vertical line with an undefined slope. The equation x = -2 indicates that every point on the line has an x-coordinate of -2. This line would pass through the point (-2, 0) on the x-axis and run parallel to the y-axis. It does not have a slope of 0 and does not represent a horizontal line.

By systematically eliminating these incorrect options, we reinforce the understanding of the distinct characteristics of horizontal and vertical lines, and the significance of the slope and y-intercept in defining a linear equation. Recognizing that x = constant represents a vertical line and y = constant represents a horizontal line is a crucial aspect of linear algebra. Additionally, understanding that the given point must satisfy the equation is essential for arriving at the correct solution.

Conclusion: The Correct Equation and Key Takeaways

Therefore, the correct equation of the line with a slope of 0 passing through the point (-2, -5) is A. y = -5. This solution highlights the importance of understanding the slope-intercept form of a linear equation and the specific characteristics of lines with a slope of 0. These lines are horizontal, have an equation of the form y = b, and maintain a constant y-value for all x-values.

Key takeaways from this problem include:

• A line with a slope of 0 is horizontal.
• The equation of a horizontal line is of the form y = b, where b is the y-intercept.
• The y-coordinate of any point on a horizontal line is constant.
• Substituting the coordinates of a point into an equation can verify if the point lies on the line.

By mastering these concepts, you can confidently tackle similar problems involving linear equations and slopes. Understanding the fundamental relationship between equations, slopes, and graphical representations is a cornerstone of success in mathematics and related fields. This problem serves as a valuable stepping stone in developing a strong foundation in linear algebra and coordinate geometry. Remember to visualize the concepts, practice with various examples, and solidify your understanding of the core principles. With consistent effort, you can confidently navigate the world of linear equations and their applications.