To solve this, we first need to understand the concept of parallel lines. Two lines are parallel if they have the same slope.

The given equation is:

$y = -\frac{3}{5}x - 7$

Here, the slope is $-\frac{3}{5}$.

Now, let’s analyze each of the multiple-choice options and convert them into slope-intercept form $y = mx + b$, where $m$ represents the slope:

1. $3x + 5y = 15$

Solve for $y$:

$5y = -3x + 15$ $y = -\frac{3}{5}x + 3$ The slope is $-\frac{3}{5}$, so this line is parallel to the given line.

2. $3y - 5x = 6$

Solve for $y$:

$3y = 5x + 6$ $y = \frac{5}{3}x + 2$ The slope is $\frac{5}{3}$, so this line is not parallel.

3. $5x + 3y = 12$

Solve for $y$:

$3y = -5x + 12$ $y = -\frac{5}{3}x + 4$ The slope is $-\frac{5}{3}$, so this line is not parallel.

4. $3x - 5y = -30$

Solve for $y$:

$-5y = -3x - 30$ $y = \frac{3}{5}x + 6$ The slope is $\frac{3}{5}$, so this line is not parallel.

Conclusion:

The equation that represents a line parallel to $y = -\frac{3}{5}x - 7$ is Option 1: $3x + 5y = 15$.

Would you like more details or have any other questions?

Here are 5 related questions for deeper understanding:

1. What is the slope-intercept form of a line?
2. How do you find the slope of a line from an equation?
3. How do you determine if two lines are perpendicular?
4. Can two lines have the same slope but not be parallel?
5. How does changing the y-intercept affect the graph of a line?

Tip: When comparing lines, focus on the slope to check for parallelism or perpendicularity.