Find the equation of the line perpendicular to the line \(3x - 2y = 6\) and passing through the point \((2,-1)\).
Answer
Simplifying this equation gives us the final equation of the line in slope-intercept form: \(y = -\frac{2}{3}x + \frac{1}{3}\).
Step 1 ：First, we rewrite the given line in slope-intercept form (\(y = mx + b\)). This gives us \(y = \frac{3}{2}x - 3\).
Step 2 ：The slope of a line perpendicular to a line with slope \(m\) is \(-\frac{1}{m}\). Thus, the slope of the line we are trying to find is \(-\frac{2}{3}\).
Step 3 ：Finally, we use the point-slope form of the equation of a line (\(y - y_1 = m(x - x_1)\)) to find the equation of the line. Plugging in the given point \((2,-1)\) and the slope \(-\frac{2}{3}\), we get \(y + 1 = -\frac{2}{3}(x - 2)\).
Step 4 ：Simplifying this equation gives us the final equation of the line in slope-intercept form: \(y = -\frac{2}{3}x + \frac{1}{3}\).
