Determining the Slope of a Line Given X-Intercept and Y-Intercept Relationship

Understanding the relationship between the x-intercept and y-intercept of a line is essential for solving geometric and algebraic problems. In this article, we will explore how to determine the slope of a line when the y-intercept is double the x-intercept.

Introduction

The slope of a line is a fundamental concept in algebra and geometry. It is defined as the change in y over the change in x, often denoted as m Δy / Δx. The equation of a line can be expressed in various forms, one of which is the intercept form. This form is particularly useful when the x-intercept and y-intercept are known.

The Problem at Hand

Consider a line whose x-intercept is (a) and y-intercept is (b). According to the problem, the y-intercept is double the x-intercept, which can be written as b 2a.

Solving for the Slope

The equation of a line in intercept form is given by:

(frac{x}{a} frac{y}{b} 1)

Substituting (b 2a), we get:

(frac{x}{a} frac{y}{2a} 1)

To eliminate the denominators, multiply the entire equation by (2a):

(2x y 2a)

Now, rearranging this equation into slope-intercept form (y mx c):

(y -2x 2a)

From this equation, the slope (m) of the line is:

(m -2)

Thus, the slope of the line is (-2).

Additional Insights

The slope of a line is determined by the relationship between the x-intercept and y-intercept. The key is to use the intercept form of the equation of a line, substitute the given relationship, and then convert it to the slope-intercept form.

For any line, the equation y mx c holds true, where m is the slope of the line and c is the y-intercept. If the y-intercept is 0, the line is parallel to the y-axis. Substituting (c 0), we get:

(y mx)

Solving for x, we get:

(frac{y}{x} m

Let's consider a more general scenario where the y-intercept is twice the x-intercept. If the x-intercept is (n eq 0) and the y-intercept is (0, 2n), then using the equation of a straight line formula:

(y - y_1 frac{y_2 - y_1}{x_2 - x_1} (x - x_1))

With (x_1 a, x_2 0, y_1 0, y_2 2a), we get:

(y - 0 frac{2a - 0}{0 - a} (x - a))

(y frac{2a}{-a} (x - a))

(y -2x - a)

(y -2x 2a)

Comparing this with the slope-intercept form y mx c, we get the slope (m -2).

Conclusion

The slope of the line is always (-2) whether the intercepts are both positive or both negative. This relationship holds true due to the consistent proportionality between the x-intercept and y-intercept.