What is the equation of the line passing through point (4, -2) that is perpendicular to the line 3x + 2y = 8?
Answer and explanation
Correct answer: D. y = (2/3)x - 14/3
- Step 1: Convert the given equation to slope-intercept form (y = mx + b): 2y = -3x + 8 => y = (-3/2)x + 4.
- Step 2: The slope of the given line is m_1 = -3/2.
- Step 3: Perpendicular lines have slopes that are negative reciprocals: m_2 = -1 / m_1 = -1 / (-3/2) = 2/3.
- Step 4: Use point-slope form with (x_1, y_1) = (4, -2): y - (-2) = (2/3)(x - 4).
- Step 5: y + 2 = (2/3)x - 8/3 => y = (2/3)x - 8/3 - 6/3 = (2/3)x - 14/3.
Answer
y = (2/3)x - 14/3.
Why not the others: 'y = (-3/2)x + 4' uses parallel slope rather than perpendicular slope; 'y = (-2/3)x + 2/3' takes the reciprocal without changing the negative sign; 'y = (2/3)x + 2' contains an arithmetic error in distributing the y-intercept.
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