Question

what is the equation of the line perpendicular to ( y = \frac{1}{2}x + 3 ) passing through the point ( (-3, 4) )?
options:
( y = \frac{1}{2}x + 5 )
( y = \frac{1}{2}x + \frac{11}{2} )
( y = -2x + 3 )
( y = -2x - 2 )

Explanation:

Step1: Find slope of perpendicular line

The given line is \( y = \frac{1}{2}x + 3 \), so its slope \( m_1=\frac{1}{2} \). For perpendicular lines, \( m_1\times m_2=-1 \), so \( m_2 = -2 \).

Step2: Use point - slope form

Point - slope form is \( y - y_1=m(x - x_1) \), with point \((-3,4)\) and \( m=-2 \). Substitute: \( y - 4=-2(x + 3) \).

Step3: Simplify the equation

Expand: \( y - 4=-2x-6 \), then add 4 to both sides: \( y=-2x - 2 \).

Answer:

\( y=-2x - 2 \) (the fourth option: \( y=-2x - 2 \))