To determine whether the given pairs of lines are parallel, perpendicular, or intersecting, we need to convert the equations into the slope-intercept form \(y = mx + b\), where \(m\) is the slope.
Let's analyze each equation one by one:
-
Equation: \(y - 6 = (x + 2)\)
Rearranging this gives: \[ y = x + 8 \] Here, the slope \(m_1 = 1\). -
Equation: \(y = -4x + 10\)
This is already in slope-intercept form. The slope \(m_2 = -4\). -
Equation: \(2x + 3y - 8 = 0\)
Rearranging this gives: \[ 3y = -2x + 8 \implies y = -\frac{2}{3}x + \frac{8}{3} \] Here, the slope \(m_3 = -\frac{2}{3}\). -
Equation: \(y = -x + 7\)
This is already in slope-intercept form. The slope \(m_4 = -1\). -
Equation: \(-2x + 3y = 12\)
Rearranging this gives: \[ 3y = 2x + 12 \implies y = \frac{2}{3}x + 4 \] Thus, the slope \(m_5 = \frac{2}{3}\).
Next, we can analyze the slopes of the lines.
- Comparing the slopes:
- Slope \(m_1 = 1\) and \(m_2 = -4\) are not equal and their product is \(-4\), indicating they are perpendicular.
- Slope \(m_3 = -\frac{2}{3}\) and \(m_5 = \frac{2}{3}\) are negative reciprocals, indicating they are perpendicular as well.
- The slopes \(m_4 = -1\) (from \(y = -x + 7\)) and \(m_3 = -\frac{2}{3}\) are not negative reciprocals, so they are not perpendicular.
- \(m_4 = -1\) and \(m_2 = -4\) are also not negative reciprocals, indicating they do not form a right angle.
In conclusion:
- Lines from pairs (1) and (2) are perpendicular.
- Lines from pairs (3) and (5) are perpendicular.
- Lines from (4) compared to (1), (2), and (3) do not add more relations of parallelism or perpendicularity since their slopes do not match or are not negative reciprocals.
Therefore, the final classification for assessing if they are parallel, perpendicular, or intersecting is:
- Parallel: None of the lines are parallel.
- Perpendicular: Lines 1 & 2, and Lines 3 & 5.
- Intersecting: All non-parallel lines (as long as they have different slopes).
So, the resulting classification among the lines based on the slopes is Perpendicular and Intersecting.
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