91Ó°ÊÓ

The slope-intercept form is one of the most common ways to express the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis. This form is especially useful because it gives you a clear understanding of the line's direction and starting point in the coordinate plane.

• The slope \( m \) tells you how steep the line is. A larger absolute value of \( m \) indicates a steeper slope.
• The y-intercept \( b \) reveals the value of \( y \) when \( x \) is zero, directly locating the line on the y-axis.

For example, the equation \( y = \frac{1}{3}x - 2 \) tells us that for every unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \), and the line crosses the y-axis at -2. This method simplifies graphing linear equations and understanding the changes in \( y \) corresponding to changes in \( x \).

The point-slope form of a line's equation is particularly useful when you know one point on the line and the slope. It is expressed as \( y - y_1 = m(x - x_1) \) where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. This form is handy for quickly drafting equations when the full line isn't immediately visible or when you need to calculate the line from limited information.

• This format highlights the changes from a particular point \((x_1, y_1)\) to any other point \((x, y)\) along the line.
• By substituting the slope and any known point into the point-slope formula, you seamlessly transition into determining the full line equation.

As in the exercise, using point (3, -1) and slope \( \frac{1}{3} \), we set up the equation \( y + 1 = \frac{1}{3}(x - 3) \). This can be further converted to slope-intercept form to make the equation more applicable for plotting or solving.

Understanding how to find the slope of a line is a fundamental skill in algebra. The slope (often denoted as \( m \)) indicates how much \( y \) changes for a change in \( x \); in simpler terms, it's the line's steepness. The formula to discover the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1}\). This calculation is straightforward if you have coordinates:

• Subtract the y-values: \( y_2 - y_1 \) in the numerator.
• Subtract the x-values: \( x_2 - x_1 \) in the denominator.
• The result gives the change in y (rise) over the change in x (run).

With the points provided in our exercise \( (3, -1) \) and \( (6, 0) \), the slope is calculated as \( \frac{0 - (-1)}{6 - 3} = \frac{1}{3} \). The positive value suggests the line ascends from left to right. Finding the slope is often the first step to produce both point-slope and slope-intercept equation forms for a line.