The problem asks us to find the equation of a line that is parallel to a given line and passes through a given point. We need to solve two such problems: 7) Find the equation of the line parallel to $x - 2y + 6 = 0$ and passing through $P(3, 0)$. 8) Find the equation of the line parallel to $x + y - 3 = 0$ and passing through $P(-2, 2)$.

1. Problem Description

The problem asks us to find the equation of a line that is parallel to a given line and passes through a given point. We need to solve two such problems:

7) Find the equation of the line parallel to $x - 2y + 6 = 0$ and passing through $P(3, 0)$.

8) Find the equation of the line parallel to $x + y - 3 = 0$ and passing through $P(-2, 2)$.

2. Solution Steps

7) The equation of the given line $L_1$ is $x - 2y + 6 = 0$. First, we find the slope of this line. We can rewrite the equation in slope-intercept form ($y = mx + b$), where $m$ is the slope.

$x - 2y + 6 = 0 \Rightarrow 2y = x + 6 \Rightarrow y = \frac{1}{2}x + 3$.

The slope of the line $L_1$ is $m_1 = \frac{1}{2}$.

Since the line we want to find is parallel to $L_1$, it must have the same slope. Thus, the slope of the new line is $m = \frac{1}{2}$.

The equation of a line with slope $m$ passing through point $(x_1, y_1)$ is given by:

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

In our case, the point is $P(3, 0)$, so $x_1 = 3$ and $y_1 = 0$. Plugging in the values, we get:

$y - 0 = \frac{1}{2}(x - 3)$

$y = \frac{1}{2}x - \frac{3}{2}$

To write it in the general form, we can multiply both sides by 2:

$2y = x - 3$

$x - 2y - 3 = 0$

8) The equation of the given line $L_1$ is $x + y - 3 = 0$. We rewrite it in slope-intercept form to find the slope.

$x + y - 3 = 0 \Rightarrow y = -x + 3$.

The slope of the line $L_1$ is $m_1 = -1$.

Since the line we want to find is parallel to $L_1$, it must have the same slope. Thus, the slope of the new line is $m = -1$.

The equation of a line with slope $m$ passing through point $(x_1, y_1)$ is given by:

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

In our case, the point is $P(-2, 2)$, so $x_1 = -2$ and $y_1 = 2$. Plugging in the values, we get:

$y - 2 = -1(x - (-2))$

$y - 2 = -1(x + 2)$

$y - 2 = -x - 2$

$y = -x$

To write it in the general form:

$x + y = 0$

3. Final Answer

7) $x - 2y - 3 = 0$

8) $x + y = 0$