How to Determine the Equation of a Line Crossing Both Axes

When faced with a scenario where a line crosses the x-axis at a specific x-coordinate and the y-axis at a particular y-coordinate, determining the equation of the line can be straightforward using the intercept form of a linear equation. This method is particularly useful for understanding the relationship between the coordinates and the slope of the line.

Intercept Form of a Line

The intercept form of the equation of a line is a powerful tool. It is given by:

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

Where (a) is the x-intercept and (b) is the y-intercept. For this problem, we are given that the line crosses the x-axis at (x 4) and the y-axis at (y -6).

Substituting the Intercepts

Substituting these values into the intercept form, the equation becomes:

(frac{x}{4} frac{y}{-6} 1) (frac{x}{4} frac{y}{-6} 1)

Converting to Slope-Intercept Form

To convert this equation into the slope-intercept form (y mx c), follow these steps:

Multiply both sides by the least common multiple (LCM) of 4 and 6, which is 12: (12 cdot left(frac{x}{4} frac{y}{-6}right) 12 cdot 1) This simplifies to: (3x - 2y 12) Rearrange the equation to solve for (y): (-2y -3x 12) (y frac{3}{2}x - 6)

Thus, the equation of the line is:

(y frac{3}{2}x - 6) (y frac{3}{2}x - 6)

Alternative Methods

There are alternative ways to determine the equation of the line based on the slope and intercepts. The slope (m) of the line can be calculated using the formula:

(m frac{Delta y}{Delta x} frac{-6 - 0}{4 - 0} frac{-6}{4} -frac{3}{2}) (m frac{Delta y}{Delta x} frac{-6 - 0}{4 - 0} frac{-6}{4} -frac{3}{2})

Using the slope-intercept form of the equation of a line (y mx c), where (c) is the y-intercept, we can substitute the values:

(y -frac{3}{2}x - 6) (y -frac{3}{2}x - 6)

However, this method results in a negative slope, which does not match the positive slope obtained in the previous method. Therefore, the correct equation of the line is:

(y frac{3}{2}x - 6) (y frac{3}{2}x - 6)

By using the intercept form and converting to the slope-intercept form, we can clearly see the relationship between the x-intercept, y-intercept, and the equation of the line.