We are given the equation of line $k$ as:

$4x + 7y = 7$

To find the equation of line $l$, which is parallel to line $k$ and passes through the point $(7, 3)$, we follow these steps:

Step 1: Find the slope of line $k$

To determine the slope of line $k$, we need to rewrite its equation in slope-intercept form, $y = mx + b$, where $m$ is the slope.

Start by solving for $y$ in the given equation:

$4x + 7y = 7$

Subtract $4x$ from both sides:

$7y = -4x + 7$

Now, divide by 7:

$y = -\frac{4}{7}x + 1$

Thus, the slope of line $k$ is $m = -\frac{4}{7}$.

Step 2: Determine the equation of line $l$

Since line $l$ is parallel to line $k$, it will have the same slope, $m = -\frac{4}{7}$. We now use the point-slope form of a line equation:

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

where $(x_1, y_1) = (7, 3)$ is the point on line $l$, and $m = -\frac{4}{7}$ is the slope.

Substitute these values into the point-slope equation:

$y - 3 = -\frac{4}{7}(x - 7)$

Step 3: Simplify the equation

Distribute the slope on the right-hand side:

$y - 3 = -\frac{4}{7}x + 4$

Now, add 3 to both sides to solve for $y$:

$y = -\frac{4}{7}x + 7$

Thus, the equation of line $l$ is:

$y = -\frac{4}{7}x + 7$

This is the equation of line $l$ in slope-intercept form.

Would you like further details or have any questions about the process? Here are some follow-up questions to explore:

1. How do you find the equation of a line given two points?
2. What changes in the equation of a line if it’s perpendicular instead of parallel?
3. How can you convert a line equation from slope-intercept form to standard form?
4. How does the slope relate to the angles between two lines?
5. What happens if the slopes of two lines are equal but they are not parallel?

Tip: When lines are parallel, they always have the same slope but different intercepts.