- Use the intercept form of the linear equation \(\frac{x}{a} + \frac{y}{b} = 1\) with the given intercepts \(a=6\) and \(b=-4\).
- Simplify the equation by multiplying by the least common multiple to obtain \(2x - 3y = 12\).
- Rearrange to the general form \(Ax + By + C = 0\), resulting in \(2x - 3y - 12 = 0\).
- Select the matching option, which is D. \(\boxed{D}\)
Explanation
- Problem Analysis
The problem asks us to determine the equation of a line that passes through the x-axis at \((6, 0)\) and the y-axis at \((0, -4)\). We are given four options in the general form \(Ax + By + C = 0\):
A\(3x - 2y - 12 = 0\)
B\(2x + 3y - 12 = 0\)
C\(2x - 3y + 12 = 0\)
D\(2x - 3y - 12 = 0\)
A linear equation can be represented in intercept form as \(\frac{x}{a} + \frac{y}{b} = 1\), where \(a\) is the x-intercept and \(b\) is the y-intercept. In this case, \(a = 6\) and \(b = -4\).
- Deriving the Equation of the Line
We start with the intercept form:
\[\frac{x}{6} + \frac{y}{-4} = 1\]
To convert this into the general form \(Ax + By + C = 0\), we first clear the denominators. The least common multiple (LCM) of 6 and 4 is 12. Multiplying the entire equation by 12 gives:
\[12 \cdot \left(\frac{x}{6}\right) + 12 \cdot \left(\frac{y}{-4}\right) = 12 \cdot 1\]
This simplifies to:
\[2x - 3y = 12\]
To write this in the form \(Ax + By + C = 0\), subtract 12 from both sides to obtain:
\[2x - 3y - 12 = 0\]
- Conclusion
Comparing our derived equation, \(2x - 3y - 12 = 0\), with the given options:
A\(3x - 2y - 12 = 0\)
B\(2x + 3y - 12 = 0\)
C\(2x - 3y + 12 = 0\)
D\(2x - 3y - 12 = 0\)
The equation corresponds to option D. Thus, the correct choice is D.
Examples
Understanding how to determine the equation of a line from its intercepts is a fundamental skill in coordinate geometry, essential for fields like engineering and architecture. For example, if an architect needs to calculate the structural beam's slope across a room defined by two points on a wall, they use this intercept method to define the line of the beam's placement accurately. Knowing the equation allows one to determine any other coordinate on that line to ensure proper alignment and support.