Question

find an equation for the line with the given properties. express your answer using either the general form or the slope - intercept form of the equation of a line. perpendicular to the line $y = -\frac{1}{2}x - 3$, containing the point $(-2, -4)$ the equation is \boxed{}. (type an equation. simplify your answer.)

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = -\frac{1}{2}x - 3 \) is \( m_1 = -\frac{1}{2} \). 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 \( (x_1,y_1)=(-2,-4) \) and \( m = 2 \). Substituting these values, we get \( y - (-4)=2(x - (-2)) \), which simplifies to \( y + 4 = 2(x + 2) \).

Step3: Simplify the equation

Expanding the right - hand side: \( y+4 = 2x + 4 \). Subtract 4 from both sides to get the slope - intercept form \( y = 2x \). Or, in general form, \( 2x - y = 0 \).

Answer:

\( y = 2x \) (or \( 2x - y = 0 \))