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Calculating the slope of a line is an essential step in determining its characteristics. The slope, represented as \( m \), measures the line's steepness. It is calculated by taking the difference in vertical positions (\( y \)-coordinates) of two points and dividing it by the difference in the horizontal positions (\( x \)-coordinates). The formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Consider two points,

• \( (-1, -2) \)
• \( (-6, -7) \)

To find the slope \( m \), plug these values into the formula: \[ m = \frac{-7 - (-2)}{-6 - (-1)} = \frac{-5}{-5} = 1 \]. This means the line rises 1 unit vertically for every unit it moves horizontally. A slope of 1 indicates a line that is inclined at a 45-degree angle, moving upwards from left to right.

Once the slope is known, we can use it to write the equation of a line in "point-slope form". This form is handy, especially when you know one point on the line and the slope. The point-slope form is: \[ y - y_1 = m(x - x_1) \] Here, \( (x_1, y_1) \) is a specific point on the line and \( m \) is the slope. In our example, using the point \((-1,-2)\) and the slope \(m = 1\), substitute into the equation: \[ y - (-2) = 1(x - (-1)) \]. When simplified, this equation transforms to: \( y + 2 = x + 1 \). The point-slope form is practical because of its ready adaptability into other forms such as the general form or slope-intercept form for further applications.

Transforming the point-slope form to the general equation form \( Ax + By = C \) involves simple algebraic manipulation. This form is widely accepted and often required for equations due to its neat, integer-only format. Starting with the point-slope derived equation: \( y + 2 = x + 1 \), subtract \( x \) from both sides to shift it to the left: \[ -x + y + 2 = 1 \]. Then subtract 2 from both sides to further simplify: \[ -x + y = -1 \]. Lastly, multiply every term by \(-1\) to remove the negative coefficient: \[ x - y = 1 \]. Now the equation is set in the general form, making it easy to interpret coefficients and constants at a glance. This final form efficiently communicates the relationship between \( x \) and \( y \) as straight integers.