Question

Write an equation of a line through \((4,5)\) that is perpendicular to \(y=\frac{1}{2} x+3\)

Short answer

The equation of the line passing through the point (4,5) and perpendicular to the line \(y=\frac{1}{2} x+3\) is \(y=-2x+13\).

Step-by-step solution

Calculate the Perpendicular Slope

The slope of the given line \(y=\frac{1}{2} x+3\) is 1/2. The slope of the line that is perpendicular to this line is the negative reciprocal of 1/2. This results in -2.

Use the Point-Slope Formula

The equation of a line can be written as \(y-y1=m(x-x1)\), where m is the slope of the line and (x1, y1) is any point on the line. Here, m is -2 and (x1, y1) is (4,5). Thus, this becomes \(y-5=-2(x-4)\).

Simplify the Equation

Distribute the -2: \(y-5=-2x+8\).

Write the Line in Slope-Intercept Form

Rearrange the equation to be in the format y=mx+b. This results in \(y=-2x+13\).