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The slope of a line is a measure of its steepness. It indicates how much the line rises or falls as you move from left to right. Calculating the slope between two points on a line involves using the slope formula, which is essential for understanding linear relationships in mathematics. By using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), the slope is determined as the ratio of the change in the vertical direction (\( y_2 - y_1 \)) to the change in the horizontal direction (\( x_2 - x_1 \)).

For the exercise, two points are given: \((-2,1)\) and \((4,-6)\). Plugging these into the slope formula gives us:

• \( x_1 = -2 \), \( y_1 = 1 \)
• \( x_2 = 4 \), \( y_2 = -6 \)

Thus, the slope \( m \) becomes \( m = \frac{-6 - 1}{4 - (-2)} = -\frac{7}{6} \). This tells us that for every 6 units the line moves horizontally, it drops by 7 units vertically.

The Point-Slope Form formula is a useful tool when you have a point on a line and the line’s slope. It allows you to write the equation of a line quickly and is given by: \( y - y_1 = m(x - x_1) \).

To apply this formula, choose one of the given points. For our example, we use \((4, -6)\). Substituting this point and the slope \(-\frac{7}{6}\) calculated in the previous section, we write:

• \( y - (-6) = -\frac{7}{6}(x - 4) \)

Simplifying this equation gives us \( y + 6 = -\frac{7}{6}x + \frac{28}{6} \). Most times, using the point closest to the origin is more intuitive, but any point on the line can be used.

The Slope-Intercept Form is perhaps the most popular form to express a linear equation, as it provides clear information about the line's slope and where it crosses the y-axis. The formula is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Continuing from the Point-Slope Form \( y + 6 = -\frac{7}{6}x + \frac{28}{6} \), we aim to express \( y \) in terms of \( x \). Rearranging gives us:

• \( y = -\frac{7}{6}x + \frac{28}{6} - 6 \)
• \( y = -\frac{7}{6}x + \frac{28}{6} - \frac{36}{6} \)
• \( y = -\frac{7}{6}x - \frac{8}{6} \)
• \( y = -\frac{7}{6}x - \frac{4}{3} \)

This final form, \( y = -\frac{7}{6}x - \frac{4}{3} \), clearly shows the line's slope \(-\frac{7}{6}\) and that it intersects the y-axis at \(-\frac{4}{3}\).