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The slope of a linear function is a key concept that describes the steepness or direction of a line on a graph. Calculating the slope allows us to understand how much a line inclines or declines as it moves across the coordinate plane. You can think of the slope as the "rise over run," where "rise" represents the change in the y-values, and "run" represents the change in the x-values.

To find the slope, use the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula tells us how for every increase of one unit in the x-direction, the y-value increases by the slope value (if positive) or decreases (if negative).
For example, given the points

• (1, 7)
• (-2, 1)

we can plug these into our formula and find the slope:

• \( \frac{1 - 7}{-2 - 1} = \frac{-6}{-3} = 2 \)

This tells us the line rises by 2 units for every 1 unit it moves to the right.

Once the slope is determined, you can use the point-slope form of a linear equation to write an equation of the line. This form is particularly useful when you know a point through which the line passes and the slope of the line.

The point-slope form is expressed as:

• \( y - y_1 = m(x - x_1) \)

"\((x_1, y_1)\)" is a point on the line, and "m" is the slope. This equation can accommodate any point on the line, maximizing its flexibility.

By substituting the point

• (1, 7)

and the previously calculated slope

• \(( m = 2 )\)

into the form, we achieve:

• \( y - 7 = 2(x - 1) \)

This is the foundational equation that describes the line's behavior at the identified point.

The slope-intercept form is one of the most accessible ways to represent a linear equation. This form emphasizes the slope and the y-intercept of the line, making it easy to interpret and graph.

The slope-intercept equation is written as:

• \( y = mx + b \)

Here, "m" stands for the slope, and "b" represents the y-intercept (the point where the line crosses the y-axis).

To convert the point-slope form \( y - 7 = 2(x - 1) \) to the slope-intercept form, we first distribute the slope:

• \( y - 7 = 2x - 2 \)

Next, add 7 to each side to isolate \( y \):

• \( y = 2x + 5 \)

This final representation reveals the line’s slope of 2 and a y-intercept at 5, providing a clear visual cue for graphing and analysis.