To find the equation of line $L_1$, which passes through the point $(3, -1)$ and is perpendicular to the line $L_2$, we'll go through the following steps:

Step 1: Determine the Slope of $L_2$

The equation of $L_2$ is: $x + 2y = 3$ To find its slope, we rewrite this in slope-intercept form ($y = mx + b$), where $m$ represents the slope.

Starting with: $x + 2y = 3$ we isolate $y$: $2y = -x + 3$ $y = -\frac{1}{2}x + \frac{3}{2}$ So, the slope of $L_2$ is $-\frac{1}{2}$.

Step 2: Find the Slope of $L_1$

Since $L_1$ is perpendicular to $L_2$, the slope of $L_1$ will be the negative reciprocal of the slope of $L_2$. Thus, the slope of $L_1$ is: $m = 2$

Step 3: Use the Point-Slope Form to Find the Equation of $L_1$

We have a point $(3, -1)$ and a slope $m = 2$. Using the point-slope form $y - y_1 = m(x - x_1)$, we get: $y - (-1) = 2(x - 3)$ $y + 1 = 2x - 6$ $y = 2x - 7$

Conclusion

The equation of $L_1$ is: $y = 2x - 7$

Would you like a further breakdown of any steps?

Here are 5 related questions that build on this information:

1. How would the equation change if $L_1$ were parallel to $L_2$ instead of perpendicular?
2. What is the slope of a line perpendicular to a line with an undefined (vertical) slope?
3. How do you determine the y-intercept of a line if you only know the slope and one point?
4. How can you verify if two lines are perpendicular by comparing their slopes?
5. What would be the equation of a line passing through $(3, -1)$ with a slope of $-2$?

Tip: Remember, when two lines are perpendicular, their slopes multiply to $-1$. This can be a quick check to confirm perpendicularity.