The problem consists of several sub-problems related to lines in a 2D plane. First, we need to find the intersection point of pairs of lines (problems 1-4). Second, we need to find the equation of a line $L_2$ that passes through a given point $P$ and is parallel to a given line $L_1$ (problems 5-8). Finally, we need to find the equation of a line $L_2$ that passes through a given point $P$ and is perpendicular to a given line $L_1$ (problems listed as 1 and 4 at the end). I will solve the 5th problem. Find the equation of a line $L_2$ that passes through $P(-1, 1)$ and is parallel to $L_1: x - y + 5 = 0$.

1. Problem Description

The problem consists of several sub-problems related to lines in a 2D plane.

First, we need to find the intersection point of pairs of lines (problems 1-4).

Second, we need to find the equation of a line $L_2$ that passes through a given point $P$ and is parallel to a given line $L_1$ (problems 5-8).

Finally, we need to find the equation of a line $L_2$ that passes through a given point $P$ and is perpendicular to a given line $L_1$ (problems listed as 1 and 4 at the end).

I will solve the 5th problem. Find the equation of a line $L_2$ that passes through $P(-1, 1)$ and is parallel to $L_1: x - y + 5 = 0$.

2. Solution Steps

To find the equation of line $L_2$, we need its slope and a point it passes through. Since $L_2$ is parallel to $L_1$, they have the same slope.

First, let's find the slope of $L_1$. We can rewrite the equation of $L_1$ in slope-intercept form ($y = mx + b$), where $m$ is the slope:

$x - y + 5 = 0$

$y = x + 5$

The slope of $L_1$ is $m_1 = 1$.

Since $L_2$ is parallel to $L_1$, its slope is also $m_2 = 1$.

Now we know the slope of $L_2$ and a point $P(-1, 1)$ that it passes through. We can use the point-slope form of a line:

$y - y_1 = m(x - x_1)$

where $(x_1, y_1)$ is the point and $m$ is the slope. Plugging in $x_1 = -1$, $y_1 = 1$, and $m = 1$:

$y - 1 = 1(x - (-1))$

$y - 1 = x + 1$

$y = x + 2$

Rewriting in the standard form:

$x - y + 2 = 0$.

3. Final Answer

The equation of the line $L_2$ is $x - y + 2 = 0$.