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The slope formula is a core concept in understanding linear functions. It helps us find the steepness of a line. The slope is represented by the letter \( m \) and calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.

The slope tells us how much \( y \) changes when \( x \) increases by one unit.

• If the slope is positive, the line rises as we move from left to right.
• If the slope is negative, the line falls.
• If the slope is zero, the line is horizontal, indicating no change in \( y \) as \( x \) changes.

For example, using the points given in the problem,

• Point 1: \((-2, 1)\)
• Point 2: \((4, -6)\)

we apply the formula to find the slope: \( m = \frac{-6 - 1}{4 - (-2)} = \frac{-7}{6} \). This tells us that the line decreases by approximately 1.167 units in \( y \) for each unit increase in \( x \).
Understanding the slope formula is crucial to forming equations for linear functions.

Once we have the slope of a line, we can use it to form the equation of the line using the point-slope form. This formula is handy. It helps us write the equation quickly when we know one point on the line and the slope. The point-slope form is represented as \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) is a point on the line and \( m \) is the slope.

For instance, in the exercise, we used the point \((-2, 1)\) and slope \(-\frac{7}{6}\) to frame the equation:

• Substitute \((-2, 1)\) and \(-\frac{7}{6}\) into the formula:

\[ y - 1 = -\frac{7}{6}(x + 2) \] This form of the equation clearly shows the relation between the specific point chosen and the slope.
The point-slope form is very useful, especially in finding equations of lines quickly without the need for complicated algebraic manipulation.

A linear equation represents a straight line in the Cartesian plane. It usually takes the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis.

In our exercise, after using the point-slope form, we continued to simplify the equation to convert it into the linear equation form.

• Start with: \( y - 1 = -\frac{7}{6}x - \frac{7}{3} \)
• Add 1 to both sides and simplify: \( y = -\frac{7}{6}x - \frac{4}{3} \)

This is the final linear equation representing the line between the points \((-2,1)\) and \((4,-6)\).
Linear equations are essential in mathematics because they form the basis for understanding more complex functions and modeling real-world situations, like predicting trends or calculating rates of change.