Answer:
To find the equation of a line that is perpendicular to the given line \(y = \frac{1}{2}x - \frac{9}{4}\) and passes through the point \((4, 1)\), we need to determine the slope of the perpendicular line first. The slope of a line perpendicular to a line with slope \(m\) is the negative reciprocal of \(m\).
Given the equation \(y = \frac{1}{2}x - \frac{9}{4}\), the slope of this line is \(m = \frac{1}{2}\). The negative reciprocal of \(\frac{1}{2}\) is \(-2\).
Now that we have the slope (\(-2\)) and the point \((4, 1)\), we can use the point-slope form of a linear equation to find the equation of the line:
\(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point and \(m\) is the slope.
Plugging in the values:
\(y - 1 = -2(x - 4)\)
Now, we can simplify this equation and write it in slope-intercept form:
\(y - 1 = -2x + 8\)
Add 1 to both sides of the equation:
\(y = -2x + 8 + 1\)
\(y = -2x + 9\)
So, the equation of the line that is perpendicular to \(y = \frac{1}{2}x - \frac{9}{4}\) and passes through the point \((4, 1)\) in slope-intercept form is \(y = -2x + 9\).