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The slope-intercept form is a way to express a linear equation of a line. In this form, the equation looks like: \[ y = mx + b \] In this equation, \( m \) represents the slope of the line. The slope indicates how steep the line is. The \( b \) represents the y-intercept, or where the line crosses the y-axis.

• slope \( m \): tells us how much \( y \) will change if \( x \) increases by 1.
• y-intercept \( b \): the y-coordinate when \( x = 0 \).

This form is very useful for quickly understanding and graphing a line. You can immediately see the direction and angle of the line just from these two parameters: \( m \) and \( b \). In our example, the line \( l: 2x + 3y - 6 = 0 \) was converted to slope-intercept form \[ y = -\frac{2}{3}x + 2 \], showing that the slope \( m \) is \(-\frac{2}{3}\), and the y-intercept \( b \) is 2.

Calculating the slope of a line is fundamental to understanding linear equations. The slope is a measure of the steepness and direction of a line. It is often denoted by \( m \), and can be calculated when you have two points on the line, as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.
In the case of line \( l \), we didn't need to calculate the slope from points, as it could be derived directly from its equation in slope-intercept form. Once we have the slope of a given line, finding the slope of a line that is perpendicular to it is simple, by taking the negative reciprocal. This means if the slope of line \( l \) is \(-\frac{2}{3}\), the slope of a line perpendicular to it will be \(\frac{3}{2}\). Calculating slopes this way is essential for solving many geometric problems about lines.

The point-slope form is another way to write the equation of a line. It is especially useful when you know a point on the line and the slope. The formula is: \[ y - y_1 = m(x - x_1) \] Where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. This form makes it easy to build the equation of a line when these conditions are met. For our task of finding the equation of a line perpendicular to \( l \) and passing through point \( P(2, 3) \), we used the point-slope form.
Given that the slope \( m = \frac{3}{2} \) and \( (x_1, y_1) = (2, 3) \), the equation becomes:\[ y - 3 = \frac{3}{2}(x - 2) \]This mathematical format gives a straightforward way to derive the full equation of a line, especially useful when point and slope information is readily available as demonstrated in finding equations of perpendicular lines.