Determining the Equation of a Horizontal Line Through Given Points

When given two points such as (6, 2) and (-4, 2), you will often encounter a situation where the points share the same y-coordinate. This indicates that the line passing through these points is horizontal. In this detailed guide, we will explore how to determine the equation of such a line using the slope-intercept form.

The Concept of a Horizontal Line

A horizontal line is a line that runs parallel to the x-axis. The defining characteristic of a horizontal line is that all its points share the same y-coordinate. This means that the y-coordinate value is constant across the entire line.

Given Points and their Significance

The points (6, 2) and (-4, 2) share the same y-coordinate (2). This tells us that for any x-value, the y-value will always be 2. Therefore, the equation of the line passing through these points is quite straightforward to determine.

Deriving the Equation

To determine the equation of a horizontal line through these points, you can use the slope-intercept form equation, (y mx b), where (m) is the slope and (b) is the y-intercept.

Step 1: Calculate the Slope

The slope (m) can be calculated using the formula:

$m frac{y_2 - y_1}{x_2 - x_1}$

For the points (6, 2) and (-4, 2), substitute the values:

$m frac{2 - 2}{-4 - 6} frac{0}{-10} 0$

This calculation shows that the slope (m) is 0, as expected for a horizontal line.

Step 2: Determine the Y-Intercept

The y-intercept (b) is the value of (y) when (x 0). However, since we are dealing with a horizontal line where every point shares the same y-coordinate, the y-intercept (b) is simply the common y-coordinate value, which is 2.

Step 3: Formulate the Equation

Now that we have the slope (m 0) and the y-intercept (b 2), we can write the equation of the line:

$y 2 implies y 2$

Therefore, the equation of the line passing through the points (6, 2) and (-4, 2) is:

$y 2$

Reiterating the Solution

The equation of the line passing through the points (6, 2) and (-4, 2) is simply:

$y 2$

This is a concise and accurate representation of the horizontal line passing through these points.

Why This Solution is Correct

By calculating the slope and y-intercept, we confirmed that the line is horizontal, with a constant y-value. The slope of zero confirms that the line is not rising or falling, and the y-intercept (the common y-value) confirms the y-coordinate for all points on the line.

Practice with a Similar Problem

Try solving with a different set of points with the same y-coordinate to further solidify your understanding. For example, consider the points (3, 5) and (9, 5).

Conclusion: The method for determining the equation of a horizontal line through given points is straightforward once you recognize the importance of the shared y-coordinate. The key steps are to calculate the slope, determine the y-intercept, and use the slope-intercept form to write the final equation.