What do the following two equations represent?
 
\[y+6 = -\dfrac{1}3\left(x-15\right)\]
\[2x+6y=24\]
Choose 1 answer:
Choose 1 answer:
(Choice A)   The same line
A
The same line
(Choice B)   Distinct parallel lines
B
Distinct parallel lines
(Choice C)   Perpendicular lines
C
Perpendicular lines
(Choice D, Checked)   Intersecting, but not perpendicular lines
D
Intersecting, but not perpendicular lines

Let's analyze the given equations to determine their relationship.

### Equation 1: \( y + 6 = -\dfrac{1}{3}(x - 15) \)
First, let's simplify and put it in slope-intercept form \(y = mx + b\):
\[
y + 6 = -\dfrac{1}{3}(x - 15)
\]
Distribute the \(-\frac{1}{3}\):
\[
y + 6 = -\dfrac{1}{3}x + 5
\]
Subtract 6 from both sides:
\[
y = -\dfrac{1}{3}x - 1
\]
So the equation is:
\[
y = -\dfrac{1}{3}x - 1
\]
This is a line with slope \( m_1 = -\dfrac{1}{3} \).

### Equation 2: \( 2x + 6y = 24 \)
Now, let's rewrite this in slope-intercept form:
\[
6y = -2x + 24
\]
Divide by 6:
\[
y = -\dfrac{1}{3}x + 4
\]
This is a line with slope \( m_2 = -\dfrac{1}{3} \).

### Analysis:
- Both lines have the same slope \( m_1 = m_2 = -\dfrac{1}{3} \).
- The y-intercepts are different: \(-1\) for the first equation and \(4\) for the second.

### Conclusion:
Since the lines have the same slope but different y-intercepts, they are **distinct parallel lines**.

So the correct answer is **Choice B: Distinct parallel lines**.

Would you like further details on this? Here are some related questions:

1. How do you find the slope of a line given an equation?
2. What is the significance of the slope and y-intercept in a linear equation?
3. How can you determine if two lines are parallel, perpendicular, or intersecting?
4. How would you graph these two lines on the coordinate plane?
5. Can you derive the point of intersection between two non-parallel lines?

**Tip:** Parallel lines never intersect and have the same slope but different y-intercepts.

What do the following two equations represent?

\[y+6 = -\dfrac{1}3\left(x-15\right)\]
\[2x+6y=24\]
Choose 1 answer:
(Choice A) The same line
(Choice B) Distinct parallel lines
(Choice C) Perpendicular lines
(Choice D, Checked) Intersecting, but not perpendicular lines

Explore how to determine the relationship between two linear equations, including identifying distinct parallel lines. This problem includes step-by-step conversion to slope-intercept form and the analysis of slopes and intercepts, suitable for students in Grades 8-10.

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