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 $y = \frac{1}{3}x - 1$; containing the point $(-3,9)$ the equation is \\(\square\\). (type an equation. simplify your answer.)

Explanation:

Step1: Find the slope of the perpendicular line

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

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 \( (x_1,y_1)=(-3,9) \) and \( m = - 3 \). Substitute these values: \( y - 9=-3(x - (-3)) \), which simplifies to \( y - 9=-3(x + 3) \).

Step3: Convert to slope - intercept form (optional, but to simplify)

Expand the right - hand side: \( y - 9=-3x - 9 \). Then add 9 to both sides: \( y=-3x - 9 + 9 \), so \( y=-3x \). We can also write it in general form as \( 3x + y = 0 \).

Answer:

The equation of the line is \( y=-3x \) (or \( 3x + y = 0 \))