Understanding the Equation of a Line Given Its X-Intercept and a Point

When dealing with the equation of a line, you often encounter different scenarios, such as finding the equation when you have the X-intercept and a point the line passes through. This article will guide you through the process of deriving the equation of a straight line given these specific pieces of information.

Identifying the X-Intercept and Point

Let's consider a line with an X-intercept of -3 units, which means it crosses the X-axis at (-3, 0). We are also given a point (3, 2) through which the line passes. Our goal is to derive the equation of this line using these details.

Using the Point-Slope Form

To find the equation, we can use the point-slope form of the equation of a line, which is given by:

y - y_1 m(x - x_1)

The slope, (m), can be calculated using the formula:

m frac{y_2 - y_1}{x_2 - x_1}

Substituting the points (-3, 0) and (3, 2) into the slope formula:

m frac{2 - 0}{3 - (-3)} frac{2}{6} frac{1}{3}

Now, using the point-slope form with the point (3, 2) and the calculated slope (m frac{1}{3}):

y - 2 frac{1}{3}(x - 3)

Rearranging this into slope-intercept form:

y - 2 frac{1}{3}x - 1

y frac{1}{3}x 1

Hence, the equation of the line is:

y frac{1}{3}x 1

Alternative Method: Using the X-Intercept and Point

Another way to find the equation is to use the intercept form of a line's equation. The intercept form is given by:

x/a y/b 1

Here, (a) is the X-intercept and (b) is the Y-intercept. Given that the X-intercept is -3, and the line passes through (3, 2), we can substitute these values into the equation:

3/-3 2/b 1

From solving this, we get:

2/b 2 Rightarrow b 1

Thus, substituting these values back into the intercept form:

x/-3 y/1 1 Rightarrow x - 3y - 3 0

Conclusion

Both methods yield the same equation: (x - 3y - 3 0). This line has an X-intercept of -3 units and passes through the point (3, 2).

Additional Resources and Further Reading

For more detailed information on lines, slopes, and equations used in geometry and algebra, refer to the following resources:

Lines, Slope, and Equations in Geometry Linear Equations and Intercept Forms