To analyze what the two equations represent, let's examine their slopes and y-intercepts.

The equations are:

1. $y + 6 = -\frac{1}{3}(x - 15)$
2. $2x + 6y = 24$

Step 1: Rewrite the equations in slope-intercept form, $y = mx + b$, where $m$ is the slope.

Equation 1:

Starting with: $y + 6 = -\frac{1}{3}(x - 15)$ Expanding and simplifying: $y + 6 = -\frac{1}{3}x + 5$ Subtracting 6 from both sides: $y = -\frac{1}{3}x - 1$

So, the slope of the first line is $m = -\frac{1}{3}$.

Equation 2:

Starting with: $2x + 6y = 24$ Solving for $y$ by first isolating $y$: $6y = -2x + 24$ Dividing by 6: $y = -\frac{1}{3}x + 4$

So, the slope of the second line is also $m = -\frac{1}{3}$.

Step 2: Determine the Relationship Between the Lines

Since both lines have the same slope ($-\frac{1}{3}$), they are parallel. However, their y-intercepts are different: -1 for the first equation and 4 for the second equation.

Conclusion

These two lines are distinct and parallel to each other.

Answer

The correct choice is: B) Distinct parallel lines

Would you like further details or have questions about any steps?

Here are five related questions to deepen your understanding:

1. How can you identify if two lines are perpendicular?
2. What happens to the graph of a line if you change only its y-intercept?
3. How can you determine if two lines are the same line?
4. What would the answer be if the slopes of the two lines were negative reciprocals of each other?
5. How do you convert a line equation to slope-intercept form if given in standard form?

Tip: Parallel lines have the same slope but different y-intercepts.