Understanding the Equation of a Line Parallel to the Y-Axis

When dealing with the equation of a line parallel to the Y-axis, we are essentially working with a vertical line. A vertical line is one that passes through all points with the same x-coordinate. Therefore, the equation of a vertical line can be expressed in the form x a, where a is the x-coordinate of any point on the line. This concept is crucial for understanding the problem of finding an equation of a line parallel to the Y-axis that passes through a given point.

Problem Definition

In the given problem, we are provided with a point (-4, -5) and a line y 1/2x - 6. Our task is to find the equation of a line that is parallel to the given line and passes through the point (-4, -5).

Step-by-Step Solution

Let's break down the solution step-by-step:

Step 1: Identifying the Slope of the Given Line

The first step in solving this problem is to identify the slope of the given line. The equation of the given line is in the slope-intercept form: y 1/2x - 6. From this form, we can see that the slope (m) of the line is 1/2.

Step 2: Using the Point-Slope Formula for the Parallel Line

The next step is to use the point-slope formula to find the equation of the line that is parallel to the given line and passes through the point (-4, -5). The point-slope formula is given by:

y - y1 m(x - x1)

Here, (x1, y1) is the given point, which is (-4, -5), and m is the slope of the given line, which is 1/2.

Substituting the values, we get:

This shows that the line parallel to the given line and passing through the point (-4, -5) has the equation y 1/2x - 3.

Another way to approach the problem is to directly determine the equation of a vertical line. Since any line parallel to the Y-axis is vertical, the equation of a vertical line passing through (-4, -5) would be:

x -4

Conclusion

By following these steps, we have successfully determined that the equation of a line parallel to the given line y 1/2x - 6 and passing through the point (-4, -5) is x -4 or y 1/2x - 3.

Understanding how to find the equation of a line parallel to another line and passing through a specific point is a fundamental concept in geometry and algebra. This knowledge can be applied to a variety of real-world scenarios, such as determining the position of vertical structures or analyzing linear relationships in data sets.