Question

1. an equation of a line perpendicular to the line represented by the equation $y = -\frac{1}{2}x - 5$ and passing through $(6, -4)$ is
2. $y = -\frac{1}{2}x + 4$
3. $y = -\frac{1}{2}x - 1$
4. $y = 2x + 14$
5. $y = 2x - 16$

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = -\frac{1}{2}x - 5 \) is \( m_1 = -\frac{1}{2} \). For two perpendicular lines, the product of their slopes is \( -1 \), i.e., \( m_1 \times m_2 = -1 \). Let the slope of the perpendicular line be \( m_2 \). So, \( -\frac{1}{2} \times m_2 = -1 \), which gives \( m_2 = 2 \).

Step2: Use point - slope form to find the equation

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

Simplify the equation:
\( y + 4 = 2x-12 \)
\( y=2x - 12 - 4 \)
\( y = 2x-16 \)

Answer:

1. \( y = 2x - 16 \)