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Slope calculation is the first step in determining the equation of a line. The slope is a measure of how steep the line is and it is calculated as the ratio of the change in the vertical direction (y-coordinates) to the change in the horizontal direction (x-coordinates). Imagine it as the amount the line "rises" for each unit it "runs".

The formula to find the slope, often represented by the letter "m," is:

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

With this formula, we simply subtract the y-coordinate of the first point from the y-coordinate of the second point and do the same for the x-coordinates. Then, divide the differences. This gives us the slope. In our exercise, using the points \((\frac{6}{5},\frac{3}{5})\) and \((\frac{1}{5},3)\), our calculations simplify to:

• First, subtract the y-values: \(3 - \frac{3}{5} = \frac{12}{5}\)
• Next, subtract the x-values: \(\frac{1}{5} - \frac{6}{5} = -1\)
• Thus, the slope \(m = \frac{\frac{12}{5}}{-1} = -\frac{12}{5}\)

Now that we have determined the slope, we can move on to finding the equation of the line.

The point-slope form is useful for directly using a single point on the line and the calculated slope to write the equation of the line. It expresses the equation in a way that highlights one point and the steepness of the line between any other points. The general structure of the point-slope form is:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) represents a point on the line, and \(m\) is the slope. We substitute the coordinates of a known point and the slope into this equation. In our scenario, using the point \((\frac{6}{5}, \frac{3}{5})\) and the calculated slope \(-\frac{12}{5}\), we have:

• \( y - \frac{3}{5} = -\frac{12}{5}(x - \frac{6}{5}) \)

This form of the equation tells us that starting from \((\frac{6}{5}, \frac{3}{5})\), for every unit we move in the x-direction, the line drops \(\frac{12}{5}\) in the y-direction. This prepares us to convert to the slope-intercept form for further simplification.

The slope-intercept form is a simple and straightforward way to express the equation of a line. In this form, the equation reveals both the slope and the y-intercept clearly, making it very intuitive. The slope-intercept form is:

• \(y = mx + b\)

Here, \(m\) is the slope, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. This form is particularly beneficial for quickly graphing the line since it gives you a starting point on the y-axis and the direction in which the line moves. To convert from point-slope to slope-intercept, we simplify:

• Substitute the slope and point into the point-slope equation: \(y - \frac{3}{5} = -\frac{12}{5}(x - \frac{6}{5})\)
• Distribute the slope and add \(\frac{3}{5}\) to both sides to get: \(y = -\frac{12}{5}x + \frac{75}{25}\)

Finally, simplifying gives us the equation in slope-intercept form:

• \( y = -\frac{12}{5}x + 3 \)

This equation shows that the line has a downward slope of \(-\frac{12}{5}\) and crosses the y-axis at 3. It provides a complete picture of the line's behavior in an easily digestible format.