- Simplify the first equation to the form \(y = -3x + 9\), identifying the slope \(m_1 = -3\).
- Simplify the second equation to the form \(y = -\frac{1}{3}x - \frac{10}{3}\), identifying the slope \(m_2 = -\frac{1}{3}\).
- Compare the slopes: they are not equal, so the lines are not parallel.
- Check the product of the slopes: \(m_1 \cdot m_2 = 1
eq -1\), so the lines are not perpendicular.
- Conclude that the relationship is \(\boxed{\text{Neither}}\).
Explanation
- Identifying the Slopes of the Lines
To determine the relationship between the two given lines, we first need to express each equation in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
For the first equation:
\[\frac{y+6x}{3} = x + 3\]
Multiply both sides by 3 to isolate \(y+6x\):
\[y + 6x = 3(x + 3)\]
\[y + 6x = 3x + 9\]
Subtract \(6x\) from both sides:
\[y = -3x + 9\]
Thus, the slope of the first line, \(m_1\), is \(-3\).
For the second equation:
\[4x - 3y = 5x + 10\]
Subtract \(4x\) from both sides to isolate the term with \(y\):
\[-3y = x + 10\]
Divide by \(-3\):
\[y = -\frac{1}{3}x - \frac{10}{3}\]
Thus, the slope of the second line, \(m_2\), is \(-\frac{1}{3}\).
- Comparing Slopes and Concluding the Relationship
Now that we have the slopes of both lines (\(m_1 = -3\) and \(m_2 = -\frac{1}{3}\)), we compare them to see if the lines are parallel or perpendicular:
- Two lines are parallel if their slopes are equal (\(m_1 = m_2\)). Comparing our slopes, \(-3
eq -\frac{1}{3}\), so the lines are not parallel.
- Two lines are perpendicular if the product of their slopes is \(-1\) (\(m_1 \cdot m_2 = -1\)). Calculating the product:
\[m_1 \cdot m_2 = (-3) \cdot \left(-\frac{1}{3}\right) = 1\]
Since the product is \(1\) and not \(-1\), the lines are not perpendicular.
Because the lines are neither parallel nor perpendicular, we conclude that they are neither.
Examples
This concept of determining whether lines are parallel or perpendicular is fundamental in fields like architecture and engineering. For example, when designing a bridge, engineers must ensure that structural supports are parallel to maintain equal load distribution or perpendicular to create rigid joints. By calculating the slopes of these supports in a coordinate plane, they can mathematically verify the alignment of the components.