Calculating the Distance Between Intercepts of the Line: -2x 5y -10

In this article, we will walk through the process to calculate the distance between the x- and y-intercepts of the line defined by the equation -2x 5y -10. Understanding these intercepts and their distance is fundamental in algebra and geometry, and it is crucial for search engine optimization (SEO) in providing valuable information to students and enthusiasts of mathematics.

Understanding Intercepts

The x- and y-intercepts are essential points in the Cartesian plane that help in visualizing and understanding a line's behavior. The x-intercept occurs where the line crosses the x-axis (y 0), and the y-intercept occurs where the line crosses the y-axis (x 0).

Step-by-Step Calculation

Let's begin by finding the x-intercept by substituting (y 0) into the equation:

Given: -2x 5y -10 Substitute (y 0): -2x 5(0) -10 Simplify: -2x -10 Solve for x: x 5 Therefore, the x-intercept is at point (5, 0).

Now, let's find the y-intercept by substituting (x 0) into the equation:

Given: -2x 5y -10 Substitute (x 0): -2(0) 5y -10 Simplify: 5y -10 Solve for y: y -2 Therefore, the y-intercept is at point (0, -2).

We have identified the intercepts at (5, 0) and (0, -2). Let's move on to calculating the distance between these intercept points.

Calculating the Distance

The distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)) is calculated using the distance formula:

[d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}]

Substituting our intercept coordinates into the formula:

[text{Distance} sqrt{(0 - 5)^2 (-2 - 0)^2}]

Let's break this down step-by-step:

(x_2 - x_1 0 - 5 -5) (y_2 - y_1 -2 - 0 -2)

Squaring these differences:

[text{Distance} sqrt{(-5)^2 (-2)^2}]

Which simplifies to:

[text{Distance} sqrt{25 4} sqrt{29}]

Thus, the distance between the x- and y-intercepts of the line -2x 5y -10 is (sqrt{29}).

Conclusion

Understanding the calculation of distances between intercepts is crucial in various mathematical applications and beyond. It provides a solid foundation for more advanced topics in analytic geometry and calculus. By mastering these calculations, students can improve their problem-solving skills and achieve better scores in mathematics exams. For SEO purposes, this article offers clear, concise instructions and includes practical examples that students can easily follow and understand.