Given: A line $Ax + By + C = 0$ and a point $(x_1, y_1)$.

To prove: The equation of the line through $(x_1, y_1)$ parallel to $Ax + By + C = 0$ is $A(x - x_1) + B(y - y_1) = 0$.

Proof:

Step 1: Find the slope of the given line.

From the general equation $Ax + By + C = 0$, we get $By = -Ax - C$.

Therefore,

$y = -\frac{A}{B}x - \frac{C}{B}$

The slope of the given line is $m = -\frac{A}{B}$.

Step 2: Use the property of parallel lines.

We know that slopes of two parallel lines are equal.

Therefore, the slope of any line parallel to $Ax + By + C = 0$ must also be $m = -\frac{A}{B}$.

Step 3: Write the equation of line through $(x_1, y_1)$ with this slope.

Using the point-slope form, the equation of a line passing through $(x_1, y_1)$ with slope $m$ is:

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

Substituting $m = -\frac{A}{B}$:

$y - y_1 = -\frac{A}{B}(x - x_1)$

Step 4: Simplify to get the required form.

Multiplying both sides by $B$:

$B(y - y_1) = -A(x - x_1)$

Rearranging:

$A(x - x_1) + B(y - y_1) = 0$

Hence proved.

Verification: The point $(x_1, y_1)$ satisfies this equation since substituting gives $A(x_1 - x_1) + B(y_1 - y_1) = 0$, which is true. Also, both lines have the same coefficients $A$ and $B$, confirming they are parallel.