91Ó°ÊÓ

The slope-intercept form of a line's equation is an easy-to-use method for representing linear equations. It appears as: \[ y = mx + b \]. Here:

• \( m \) is the slope of the line, indicating the steepness and direction.
• \( b \) is the y-intercept, where the line crosses the y-axis.

This form is particularly useful because it provides direct insights into the properties of the line. For example, consider the equation \( y = \frac{2}{9}x + \frac{13}{9} \). Here, the slope \( m = \frac{2}{9} \) tells us the line rises \(2\) units for every \(9\) units it moves horizontally. The y-intercept \( b = \frac{13}{9} \) indicates the point at which the line crosses the y-axis.

The point-slope form is another way to write the equation of a line, especially handy when you have a known slope and a specific point it passes through. It is expressed as: \[ y - y_1 = m(x - x_1) \]. Here, \( (x_1, y_1) \) is a known point on the line, and \( m \) is the slope. This form is often the stepping stone to deriving other forms like slope-intercept. For example, using the point \((-2, 1)\) and a slope of \( \frac{2}{9} \), we have: \[ y - 1 = \frac{2}{9}(x + 2) \]. This formula can be rearranged to yield the slope-intercept form, making it very versatile in different scenarios.

Perpendicular lines intersect at right angles, which means their slopes are negative reciprocals of each other. To find a line perpendicular to a given line, calculate this reciprocal. For example, consider the line represented by \( 3x + 4y = 9 \). To find a perpendicular line, first determine its slope: rewrite it as \( y = -\frac{3}{4}x + \frac{9}{4} \). The negative reciprocal of \(-\frac{3}{4}\) is \( \frac{4}{3} \). Thus, any line with a slope of \( \frac{4}{3} \) is perpendicular to the given line. However, if the original line is horizontal, like \( y = 4 \), then the perpendicular line will be vertical, which means it has an undefined slope. For example, a line passing through \((-2, 1)\) perpendicular to \( y = 4 \) would be \( x = -2 \).

Parallel lines never intersect and thus have identical slopes. If you need to find a line parallel to another, just ensure they share the same slope. Take, for instance, the line \( 3x - 2y = 5 \). When rearranged into the slope-intercept form, it becomes \( y = \frac{3}{2}x - \frac{5}{2} \), revealing a slope of \( \frac{3}{2} \). A line parallel to this would also have a slope of \( \frac{3}{2} \). For a practical application, if such a line needs to pass through a specific point like \((-2, 1)\), use the point-slope form: \[ y - 1 = \frac{3}{2}(x + 2) \]. This method quickly gives you the desired parallel line equation.