Consider: $y=\frac{3}{2} x+\frac{1}{4}$
Find the equation of the line perpendicular to this line and passes through the point $(1,-2)$

Step 1 ：The given line equation is \(y = \frac{3}{2}x + \frac{1}{4}\). The slope of this line is \(\frac{3}{2}\).

Step 2 ：The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Therefore, the slope of the line perpendicular to the given line is \(-\frac{2}{3}\).

Step 3 ：We know that the equation of a line is given by \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. We know the slope and a point on the line, so we can substitute these values into the equation to find the y-intercept.

Step 4 ：Substituting the slope \(m = -\frac{2}{3}\) and the point \((1, -2)\) into the equation, we get \(-2 = -\frac{2}{3} * 1 + c\). Solving for \(c\), we get \(c = -\frac{4}{3}\).

Step 5 ：Now that we have the slope and y-intercept, we can write the equation of the line.

Step 6 ：\(\boxed{y = -\frac{2}{3}x - \frac{4}{3}}\) is the equation of the line that is perpendicular to the given line and passes through the point \((1, -2)\).