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The slope formula is a key concept in understanding linear functions. It helps you determine the steepness or inclination of a line.
The slope is calculated by taking the difference in the y-values (up and down) over the difference in the x-values (side to side) for two given points on a line. The formula is written as:

• \( m = \frac{y_2-y_1}{x_2-x_1} \)

This formula tells you how much y changes for a change in x. If the result is positive, the line moves upwards, and if it's negative, it moves downwards.
In our exercise, we used points (3, -2) and (-1, -4), substituted them into the formula, and found that the slope \( m \) is \( \frac{1}{2} \). This means that for every 2 units the line moves horizontally, it moves 1 unit up vertically.

The point-slope form is an essential part of finding the equation of a line. Once you have the slope, you can use the point-slope formula when you know a point on the line and the slope. It is written as:

• \( y - y_1 = m(x - x_1) \)

In this formula, \((x_1, y_1)\) is a known point, and \(m\) is the slope you calculated.
For the given exercise, using the point (3, -2) and a slope of \( \frac{1}{2} \), we plug these values into the formula:

• \( y + 2 = \frac{1}{2}(x - 3) \)

After performing basic algebraic operations, we simplify this into an equation. This form is handy because it clearly shows the slope and a specific point on the line.

The slope-intercept form is one of the most popular formats for linear equations. It makes it easy to interpret and graph a line. Written as:

• \( y = mx + b \)

Here, \(m\) represents the slope, and \(b\) is the y-intercept, which is where the line crosses the y-axis.
After simplifying the equation from the point-slope form, we got it into the slope-intercept form:

• \( y = \frac{1}{2}x - \frac{7}{2} \)

This final equation allows you to quickly see both the slope and where the line crosses the y-axis. It's very useful for graphing and understanding how the line behaves in relation to the axes.