First, let's find the slope of the given line. We can rewrite the equation in slope-intercept form (y = mx + b) by subtracting 3 from both sides of the equation:
y - 3 = (8/3)(x + 2)
y = (8/3)(x + 2) + 3
y = (8/3)(x) + (8/3)(2) + 3
y = (8/3)x + 16/3 + 3
y = (8/3)x + 16/3 + 9/3
y = (8/3)x + 25/3
From this equation, we can see that the slope of the given line is 8/3.
To find the slope of a line perpendicular to the given line, we take the negative reciprocal of the slope. So the perpendicular line will have a slope of -3/8.
Next, we can use the point-slope form of a linear equation (y - y₁ = m(x - x₁)) to find the equation of the line passing through the point (-2, 3) with a slope of -3/8:
y - 3 = (-3/8)(x - (-2))
y - 3 = (-3/8)(x + 2)
y - 3 = (-3/8)x - 3/4
y = (-3/8)x - 3/4 + 3
y = (-3/8)x - 3/4 + 12/4
y = (-3/8)x + 9/4
Therefore, the equation of the line perpendicular to the given line and passing through the point (-2, 3) is y = (-3/8)x + 9/4.
The correct answer is D.
Write the equation of a line that is perpendicular to the given line and that passes through the given point.
y - 3 = 8/3 (x + 2); (-2, 3)
A. y + 3 = -3/8 (x - 2)
B. y - 3 = -3/8 (x + 2)
C. y - 2 = -3/8 (x + 3)
D. y + 3 = 3/8 (x - 2)
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