Question

use the given conditions to write an equation for the line in point - slope form and in slope - intercept form. passing through (5, - 3) and perpendicular to the line whose equation is $y=\frac{1}{2}x + 3$ write an equation for the line in point - slope form. (simplify your answer. use integers or fractions for any numbers in the equation.)

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = \frac{1}{2}x + 3 \) is \( m_1=\frac{1}{2} \). For two perpendicular lines, the product of their slopes is -1, so \( m_1\times m_2=-1 \). Solving for \( m_2 \), we get \( m_2 = - \frac{1}{m_1} = - \frac{1}{\frac{1}{2}} = -2 \).

Step2: Use point - slope form formula

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(5,-3) \) and \( m=-2 \). Substituting these values into the formula, we have \( y - (-3)=-2(x - 5) \), which simplifies to \( y + 3=-2(x - 5) \).

Answer:

\( y + 3=-2(x - 5) \)