Find an equation of the line perpendicular to $2 x-3 y=6$ that passes through the point (4, -1).
We are given the line equation \(2x - 3y = 6\) and we need to find the equation of a line that is perpendicular to it and passes through the point \((4, -1)\).
First, we rewrite the given line equation in slope-intercept form, \(y = mx + b\), to find its slope:
\[2x - 3y = 6\]
Subtract \(2x\) from both sides:
\[-3y = -2x + 6\]
Divide by \(-3\):
\[y = \frac{2}{3}x - 2\]
So, the slope of the given line, \(m_1\), is \(\frac{2}{3}\).
Two lines are perpendicular if the product of their slopes is \(-1\). Let \(m_2\) be the slope of the perpendicular line. Then, we have:
\[m_1 \cdot m_2 = -1\]
\[\frac{2}{3} \cdot m_2 = -1\]
Solving for \(m_2\):
\[m_2 = -1 \cdot \frac{3}{2} = -\frac{3}{2}\]
Thus, the slope of the perpendicular line is \(-\frac{3}{2}\).
Now, we use the point-slope form of a linear equation, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (4, -1)\) and the slope \(m = -\frac{3}{2}\):
\[y - (-1) = -\frac{3}{2}(x - 4)\]
\[y + 1 = -\frac{3}{2}x + 6\]
To express this in a standard form, multiply every term by \(2\):
\[2(y + 1) = -3(x - 4)\]
\[2y + 2 = -3x + 12\]
Rearrange the terms by adding \(3x\) to both sides and subtracting \(2\) from both sides:
\[3x + 2y = 10\]
This matches option (C).
Imagine designing the layout for a new urban street grid. Engineers must ensure that intersecting streets meet at right angles (perpendicularly) to optimize traffic flow and structural integrity. By calculating the slopes of existing roads, they can determine the precise path and equation for new roads to intersect them correctly, much like finding the equation of the perpendicular line in this problem. Understanding these geometric relationships is fundamental in civil engineering, architecture, and urban planning, ensuring cities are built with precision and efficiency.