Finding the Equation of a Line Given a Point and X-Intercept

In this article, we explore the process of finding the equation of a line that passes through a specific point and has a given x-intercept. This is a common task in mathematics, often encountered in high school algebra and beyond. We will walk through the step-by-step process using a concrete example.

Introduction to Line Equations

A line in the Cartesian plane can be described using various forms of equations. The most common forms are the slope-intercept form (y mx b), the point-slope form (y - y_1 m(x - x_1)), and the intercept form (frac{x}{a} frac{y}{b} 1), where (a) and (b) are the x-intercept and y-intercept, respectively.

Problem: Finding the Equation of a Line

Given: A line passes through the point ((-4, 6)) and has an x-intercept of (8) units.

Step 1: Identify the X-Intercept

The x-intercept is the point where the line crosses the x-axis, which means (y 0). Therefore, the x-intercept point is ((8,0)).

Step 2: Determine the Slope

The slope (m) of the line can be calculated using the formula: [m frac{y_2 - y_1}{x_2 - x_1}] Here, we use the points ((-4, 6)) and ((8, 0)): [m frac{0 - 6}{8 - (-4)} frac{-6}{12} -frac{1}{2}]

Step 3: Use the Point-Slope Form

The point-slope form of a line's equation is given by: [y - y_1 m(x - x_1)] Using point ((-4, 6)) and the slope (-frac{1}{2}): [y - 6 -frac{1}{2}(x 4)] Simplifying: [y - 6 -frac{1}{2}x - 2] [y -frac{1}{2}x 4]

Conclusion

The equation of the line in slope-intercept form is: [boxed{y -frac{1}{2}x 4}]

Alternative Approach: Using Intercept Form

Alternatively, we can express the line in intercept form (frac{x}{a} frac{y}{b} 1), where (a) is the x-intercept and (b) is the y-intercept. Given (a 8), the equation becomes: [frac{x}{8} frac{y}{b} 1] Since the point ((-4, 6)) lies on the line, we substitute it into the equation: [frac{-4}{8} frac{6}{b} 1] [-frac{1}{2} frac{6}{b} 1] [frac{6}{b} frac{3}{2}] [b 4] Thus, the equation of the line in intercept form is: [frac{x}{8} frac{y}{4} 1] Multiplying both sides by 8, we get the equation in standard form: [x 2y 8]

Both methods yield the same line equation, regardless of the form used. The choice of method depends on the specific problem and the form in which the final answer is requested.

Conclusion

By following these steps, we can determine the equation of a line given a point and the x-intercept. Whether using the slope-intercept form or the intercept form, both methods provide a clear and concise solution to the problem.

Key Takeaways: - slope-intercept form - x-intercept - point-slope form