Question

find the equation of the line that is perpendicular to $y = -\frac{1}{2}x + 1$ and contains the point (6,17).
$y = \boxed{?}x + \boxed{}$

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = -\frac{1}{2}x + 1 \) is \( m_1 = -\frac{1}{2} \). The slope of a line perpendicular to it, \( m_2 \), satisfies \( m_1 \times m_2 = -1 \). 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 \( (x_1,y_1)=(6,17) \) and \( m = 2 \). Substituting these values, we get \( y - 17 = 2(x - 6) \).

Step3: Simplify the equation

Expand the right - hand side: \( y - 17 = 2x-12 \). Then, add 17 to both sides: \( y=2x + 5 \).

Answer:

\( y = 2x+5 \)