Understanding the X-Intercept of a Line Through Points

The x-intercept of a line is a crucial concept in algebra and geometry, representing the point where a line crosses the x-axis. This tutorial will guide you through the process of finding the x-intercept of a line passing through two specific points, 1, 4 and 4, 1. This topic is essential for both beginners and those looking to refresh their skills in coordinate geometry.

What is the X-Intercept for a Line Passing Through 1, 4 and 4, 1?

The question asks for the x-intercept of a line passing through the points 1, 4 and 4, 1. The answer is not 4.5, as some might confuse it with the y-intercept. Let’s explore how to accurately find the x-intercept using a systematic approach.

Calculating the Slope of the Line

First, we need to determine the slope of the line that passes through the given points. The formula for the slope is:

[math]m frac{y_2 - y_1}{x_2 - x_1}[/math]

Given the points (1, 4) and (4, 1), we can substitute these values into the formula:

[math]m frac{1 - 4}{4 - 1} frac{-3}{3} -1[/math]

Formulating the Equation of the Line

With the slope known, we can use the point-slope form of the equation of a line:

[math]y - y_1 m(x - x_1)[/math]

Let’s use the point (1, 4) to create the equation:

[math]y - 4 -1(x - 1)[/math]

Simplifying this, we get:

[math]y - 4 -x 1[/math]

[math]y -x 5[/math]

Finding the X-Intercept

The x-intercept is the point where the line crosses the x-axis, meaning the y-coordinate is 0. To find the x-intercept, we set y to 0 in the equation:

[math]0 -x 5[/math]

Solving for x:

[math]x 5[/math]

Therefore, the x-intercept is the point (5, 0).

Conclusion and Recap

By following these steps, we have accurately determined that the x-intercept of the line passing through the points (1, 4) and (4, 1) is 5. This process involves calculating the slope, finding the line equation, and then identifying the x-intercept by setting y to zero.

Related Questions

If you have other questions about finding the x-intercept or understanding the equation of a line, feel free to ask. Our goal is to help you master these important mathematical concepts.

Additional Reading

For a deeper dive into coordinate geometry, consider exploring the following resources: