91Ó°ÊÓ

The slope-intercept form of a line is one of the most recognizable equations in algebra. It is expressed as \( y = mx + c \), where:

• \( m \) represents the slope of the line, which dictates its steepness.
• \( c \) is the y-intercept, the point where the line crosses the y-axis.

This form is particularly handy because it provides both the slope and the y-intercept directly. You can readily identify how the line behaves just by looking at the equation. For example, in the slope-intercept form \( y = 2x - 1 \), we can tell that the line rises by 2 units on the y-axis for every 1 unit it moves to the right on the x-axis, and it crosses the y-axis at \( y = -1 \). This makes it quite straightforward to sketch the line or compare it to others.

Understanding how to calculate the slope is crucial for grasping the behavior of a line. The slope is essentially the rate of change, or steepness, of a line, often referred to as \( m \). You can calculate it using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line. In our exercise, it involves the points \((3,5)\) and \((8,15)\). Plugging into the formula, we get \( m = \frac{15 - 5}{8 - 3} = \frac{10}{5} = 2 \). Thus, the slope \( m = 2 \) indicates that for every unit the line moves right, it climbs 2 units upward. With the slope, one gets insights into how the line is oriented in space.

Creating the equation of a line involves knowing at least two critical characteristics: the slope and a point on the line. Armed with this info, you can use the point-slope form, which is \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope, and \( (x_1, y_1) \) is a specific point that the line passes through. To derive the equation from our given line passing through \((3,5)\) with a slope of 2, substitute these values into the point-slope equation: \( y - 5 = 2(x - 3) \). This equation succinctly represents the line's behavior. If we want the more familiar slope-intercept form, expand the equation to isolate \( y \). Thus, \( y - 5 = 2x - 6 \), and with a bit of rearranging, you get \( y = 2x - 1 \). Whether using point-slope or slope-intercept, forming the equation of a line is key in predicting and visualizing linear relationships.