To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we can convert the equations into the slope-intercept form \(y = mx + b\) where \(m\) is the slope.
Pair 1
-
First Line:
\(y - 6 = 14(x + 2)\)
Expand:
\(y - 6 = 14x + 28\)
Therefore,
\(y = 14x + 34\)
Slope: \(m_1 = 14\) -
Second Line:
\(y = -4x - 10\)
Slope: \(m_2 = -4\)
Analysis of Pair 1
- The slopes are \(m_1 = 14\) and \(m_2 = -4\).
- Since the product of the slopes \(m_1 \cdot m_2 = 14 \cdot (-4) = -56\), which is negative, the lines are perpendicular.
Pair 2
-
First Line:
\(2x - y = 7\)
Rearranging gives:
\(y = 2x - 7\)
Slope: \(m_1 = 2\) -
Second Line:
\(3x - 4y = -8\)
Rearranging gives:
\(4y = 3x + 8 \implies y = \frac{3}{4}x + 2\)
Slope: \(m_2 = \frac{3}{4}\)
Analysis of Pair 2
- The slopes are \(m_1 = 2\) and \(m_2 = \frac{3}{4}\).
- Since the product of the slopes \(m_1 \cdot m_2 = 2 \cdot \frac{3}{4} = \frac{3}{2}\), which is positive, the lines are intersecting.
Pair 3
-
First Line:
\(y = -23x + 7\)
Slope: \(m_1 = -23\) -
Second Line:
\(-2x - 3y = 12\)
Rearranging gives:
\(-3y = 2x + 12 \implies y = -\frac{2}{3}x - 4\)
Slope: \(m_2 = -\frac{2}{3}\)
Analysis of Pair 3
- The slopes are \(m_1 = -23\) and \(m_2 = -\frac{2}{3}\).
- Since the slopes have the same sign and are not equal (not parallel), the lines are intersecting.
Summary of the results:
- Pair 1: Perpendicular
- Pair 2: Intersecting
- Pair 3: Intersecting