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The slope of a line is a key component in understanding linear functions. It represents the rate at which the y-value changes for every unit increase in the x-value. This concept is crucially tied to the inclination of the line on a graph.

To find the slope between two points, \(x_1, y_1\) and \(x_2, y_2\), we use the formula:

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

This formula gives you the 'rise over run', illustrating how much the line rises vertically for each horizontal step.

For instance, using the points \(2,4\) and \(3,9\) from the exercise, the slope is calculated as:

• \( m = \frac{9 - 4}{3 - 2} = 5 \)

A slope of 5 indicates that the line increases by 5 units in the y-direction for every unit it moves to the right in the x-direction. This tells us how steep the line is and determines its direction.

Once you have the slope, you can write the equation of the line using the point-slope form. This form is useful because it allows you to build a line equation directly from a known point and the slope.

The point-slope form of a linear equation is:

• \( y - y_1 = m(x - x_1) \)

Here, \(m\) is the slope, and \(x_1, y_1\) are the coordinates of a known point on the line.

For the example with points (2,4) and slope 5, choosing point (2,4) gives:

• \( y - 4 = 5(x - 2) \)

This equation captures the line's characteristics based on just one point and its slope. It's a straightforward way to express linear relationships when you're given specific points.

The slope-intercept form of a line is very popular because of its simplicity. It's straightforward to interpret and easily identifies the slope and y-intercept directly from its structure.

The slope-intercept equation is structured as:

• \( y = mx + b \)

Here, \(m\) is the slope and \(b\) is the y-intercept, indicating where the line crosses the y-axis.

Converting the given point-slope form \( y - 4 = 5(x - 2) \) into slope-intercept form involves distributing and simplifying:

• Distribute 5: \( y - 4 = 5x - 10 \)
• Add 4 to both sides: \( y = 5x - 6 \)

Thus, \( y = 5x - 6 \) is the slope-intercept form. Here, the equation format makes it easy to see the line's slope of 5 and a y-intercept of -6, providing a complete picture of the line's behavior on a graph.