Question

question
find the equation of a line perpendicular to $y = -\frac{1}{2}x + 9$ that passes through the point $(1,7)$.

answer
\\(\circ\\) $y + 7 = -\frac{1}{2}(x + 1)$
\\(\circ\\) $y - 7 = \frac{1}{2}(x - 1)$
\\(\circ\\) $y - 7 = 2(x - 1)$
\\(\circ\\) $y - 7 = 2(x + 1)$

Explanation:

Step1: Find slope of perpendicular line

The given line is \( y = -\frac{1}{2}x + 9 \), so its slope \( m_1 = -\frac{1}{2} \). For perpendicular lines, \( m_1 \times m_2 = -1 \). Solving \( -\frac{1}{2} \times m_2 = -1 \), we get \( m_2 = 2 \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(1,7) \) and \( m = 2 \). Substituting these values, we get \( y - 7 = 2(x - 1) \).

Answer:

\( y - 7 = 2(x - 1) \) (the option with this equation)