Question

3x-y=5
3x+6y=24
parallel


perpendicular


intersecting but not perpendicular


None of the above.

Answer 1

To determine the relationship between the two lines given by the equations \(3x - y = 5\) and \(3x + 6y = 24\), we can convert both equations into slope-intercept form \(y = mx + b\), where \(m\) is the slope.

1. **First equation: \(3x - y = 5\)**

Rearranging gives:
\[
-y = -3x + 5
\]
Multiplying through by -1:
\[
y = 3x - 5
\]
The slope \(m_1\) of the first line is \(3\).

2. **Second equation: \(3x + 6y = 24\)**

Rearranging gives:
\[
6y = -3x + 24
\]
Dividing by 6:
\[
y = -\frac{1}{2}x + 4
\]
The slope \(m_2\) of the second line is \(-\frac{1}{2}\).

Now, we can analyze the slopes:

- If the slopes are equal (\(m_1 = m_2\)), the lines are **parallel**.
- If the product of the slopes is \(-1\) (\(m_1 \cdot m_2 = -1\)), the lines are **perpendicular**.
- If the slopes are neither equal nor the product of the slopes equals \(-1\), the lines are **intersecting but not perpendicular**.

Here, we have:
- \(m_1 = 3\)
- \(m_2 = -\frac{1}{2}\)

Calculating the product:
\[
m_1 \cdot m_2 = 3 \cdot \left(-\frac{1}{2}\right) = -\frac{3}{2} \quad (\text{not } -1)
\]

Since \(m_1\) and \(m_2\) are not equal and the product is not -1, the lines are intersecting but not perpendicular.

**Final Answer: Intersecting but not perpendicular.**