We are tasked with solving the given problem. Here's the step-by-step solution:

Problem Analysis

1. Equation of $L_1$: $y = 6 - 2x$

   • Slope ($m_1$) = $-2$.

2. Line $L_2$:

   • Perpendicular to $L_1$: This means its slope $m_2$ satisfies $m_1 \cdot m_2 = -1$.
   • Passes through the point $(4, 7)$.
   • Find where $L_2$ crosses the x-axis.

Step 1: Determine Slope of $L_2$

If $m_1 = -2$, then the slope of $L_2$ is: $m_2 = -\frac{1}{m_1} = -\frac{1}{-2} = \frac{1}{2}.$

So, the slope of $L_2$ is $\frac{1}{2}$.

Step 2: Equation of ( L_2 \