Questions: Use the given conditions to write an equation for the line in point-slope form and in slope Passing through (6,-8) and perpendicular to the line whose equation is y=1/3 x+2 Write an equation for the line in point-slope form. y+8=1/3(x-6) (Simplify your answer. Use integers or fractions for any numbers in the equation.) Write an equation for the line in slope-intercept form. (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Solution Steps

To solve this problem, we need to find the equation of a line that passes through the point (6, -8) and is perpendicular to the line given by the equation \( y = \frac{1}{3}x + 2 \).

1. Find the slope of the given line: The slope of the line \( y = \frac{1}{3}x + 2 \) is \( \frac{1}{3} \).
2. Determine the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is \( -3 \).
3. Use the point-slope form: The point-slope form of a line is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Plug in the point (6, -8) and the slope \( -3 \) to get the equation in point-slope form.
4. Convert to slope-intercept form: Simplify the point-slope form equation to get the slope-intercept form \( y = mx + b \).

The slope of the given line \( y = \frac{1}{3}x + 2 \) is \( \frac{1}{3} \). The slope of a line perpendicular to this line is the negative reciprocal of \( \frac{1}{3} \), which is \( -3 \).

The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Substituting \( m = -3 \) and the point \( (6, -8) \), we get: \[ y - (-8) = -3(x - 6) \] Simplifying, we have: \[ y + 8 = -3(x - 6) \]

To convert the point-slope form to slope-intercept form \( y = mx + b \), we simplify the equation: \[ y + 8 = -3(x - 6) \] \[ y + 8 = -3x + 18 \] \[ y = -3x + 18 - 8 \] \[ y = -3x + 10 \]

Final Answer