Question

perpendicular line
flip the given slope and change sign
b. write the equation of a line that is perpendicular to ( y = 2x + 1 ) and goes through the point ( (-6, 4) )
( f = \frac{1}{2} ) ( y = 2x + 1 )
( m = -\frac{1}{2} ) ( y - 4 = -\frac{1}{2}(x + 6) )
( (x_1, y_1) )
( y_1 = 4 )

Explanation:

Step1: Find the slope of the perpendicular line

The given line is \( y = 2x + 1 \), so its slope \( m_1 = 2 \). For a line perpendicular to it, the slope \( m_2 \) satisfies \( m_1 \times m_2=- 1 \). So \( m_2=-\frac{1}{2} \) (by flipping the slope \( \frac{2}{1} \) to \( \frac{1}{2} \) and changing the sign).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(-6,4) \) and \( m =-\frac{1}{2} \). Substituting these values, we get \( y - 4=-\frac{1}{2}(x + 6) \).

Step3: Convert to slope - intercept form (optional, for a more standard form)

Expand the right - hand side: \( y-4 =-\frac{1}{2}x-3 \). Then add 4 to both sides: \( y=-\frac{1}{2}x + 1 \).

Answer:

The equation of the line is \( y=-\frac{1}{2}x + 1 \) (or in point - slope form \( y - 4=-\frac{1}{2}(x + 6) \))