Determining Parallel and Perpendicular Lines

What can be determined from the slopes of the given lines?

The slopes of the given lines are -3 and [tex]\\( \\frac{1}{3} \\)[/tex]. Determine if the lines are parallel, perpendicular, or neither.

Analysis of Slopes

Given the slopes of -3 and [tex]\\( \\frac{1}{3} \\)[/tex], it is possible to determine the relationship between the lines.

To determine if the lines are parallel or perpendicular, we need to assess the slopes of each line. The slope-intercept form of an equation allows us to easily identify the slope by looking at the coefficient of x.

The slope of the first line, y = -3x + 6, is -3. This indicates a steep downward slope. For the second line, y = [tex]\\( \\frac{1}{3} \\)x - 8[/tex], the slope is [tex]\\( \\frac{1}{3} \\)[/tex], representing a less steep upward slope.

When comparing the slopes, we find that they are negative inverses of each other. The product of the slopes is -1, indicating that the lines are perpendicular. Therefore, the equations y = -3x + 6 and y = [tex]\\( \\frac{1}{3} \\)x - 8[/tex] represent perpendicular lines.