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Calculating the slope of a line is an essential skill in algebra that tells us how steep the line is. The slope, often represented by \( m \), measures the change in the vertical position (the rise) for a unit change in the horizontal position (the run) between two points. To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Substituting in the points \((3, 5)\) and \((8, 15)\), we calculate:

• Rise = \( 15 - 5 = 10 \)
• Run = \( 8 - 3 = 5 \)
• Slope \( m = \frac{10}{5} = 2 \)

The result \( m = 2 \) tells us that for each step to the right along the x-axis, the line goes 2 steps up.

The slope-intercept form of a line is one of the most common ways to express the equation of a line. It emphasizes the slope and the y-intercept, which is where the line crosses the y-axis. The formula for the slope-intercept form is:

• \( y = mx + b \)

Here, \( m \) represents the slope, and \( b \) represents the y-intercept. Knowing these two values allows us to understand and sketch the line quickly on a graph.
For our problem, we must transform the point-slope form equation we derived into the slope-intercept form. Beginning with the equation \( y - 5 = 2(x - 3) \), we simplify it:

• First, expand: \( y - 5 = 2x - 6 \)
• Then, add 5 to both sides to solve for \( y \):
  \( y = 2x - 6 + 5 \)
• This simplifies to \( y = 2x - 1 \)

In the final equation \( y = 2x - 1 \), 2 is the slope and -1 is the y-intercept. This shows the line crosses the y-axis at \((0, -1)\).

An equation of a line provides a complete description of the line’s path across a graph. There are different forms, but the two most commonly used are the point-slope form and the slope-intercept form.
The point-slope form is particularly useful when you know a point on the line and the slope. The formula is:

• \( y - y_1 = m(x - x_1) \)

Using a known point \((3, 5)\) and our calculated slope \(2\), we get: \( y - 5 = 2(x - 3) \). This form tells us the slope and a specific point the line passes through, giving us a precise way to sketch a line.
On the other hand, the slope-intercept form \( y = mx + b \) puts emphasis on the overall slope of the line and where it crosses the y-axis. Both forms give critical insights into the line's behavior and how it fits within a graph.