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The slope of a line is a measure of its steepness and direction. It represents how much the line rises or falls as you move from left to right. Mathematically, the slope (\(m\)) is calculated by the change in the y-coordinates divided by the change in the x-coordinates between two points on the line. This is sometimes called "rise over run."

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

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

A positive slope means the line is ascending as you move from left to right, whereas a negative slope means it is descending. If the slope is zero, the line is horizontal, indicating no vertical change. Conversely, an undefined slope indicates a vertical line, where there is no horizontal change.

Understanding the slope is crucial because it helps us understand the behavior of the line just by looking at the equation. In our problem, the slope of \(-\frac{1}{3}\) indicates a gentle decrease from left to right.

The y-intercept is a key aspect of a linear equation as it indicates where the line crosses the y-axis. In other words, it's the value of \(y\) when \(x = 0\). This value can be found by rearranging the linear equation into the format \( y = mx + c \), where \(c\) is the y-intercept.

To find the y-intercept, you can substitute the slope and coordinates of one point back into the equation and solve for \(c\). For example, using the point\((-4, 3)\)and knowing our slope is \(-\frac{1}{3}\), we substitute these into the equation \(y = mx + c\) as follows:

• \(3 = -\frac{1}{3}(-4) + c\)

Solving this equation allows us to find the y-intercept as \(\frac{5}{3}\). This step is essential because once we have the y-intercept, plotting the line on a graph becomes much easier.

The equation of a line is fundamentally important in describing the relationship between the x and y coordinates in a two-dimensional space. The most common form of this equation is the slope-intercept form: \( y = mx + c \), where \(m\) is the slope and \(c\) is the y-intercept.

Once we know the slope and y-intercept, constructing the equation is straightforward. You simply plug these values into the equation. For our problem, we've calculated the slope as \(-\frac{1}{3}\) and the y-intercept as \(\frac{5}{3}\). Placing these into the slope-intercept form gives us:

• \( y = -\frac{1}{3}x + \frac{5}{3} \)

With this equation, you can predict the y-value for any given x-value that falls on the line, making it a powerful tool for analysis and understanding linear relationships.