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The slope of a line measures its steepness and direction. It is often symbolized by the letter m.
The slope can be calculated as the change in the y values divided by the change in the x values between two points on the line: \[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \].
A positive slope means the line rises as it moves from left to right, and a negative slope means the line falls.
A slope of zero indicates a horizontal line, and an undefined slope corresponds to a vertical line.
Understanding these core ideas about the slope helps to analyze and interpret the behavior of linear functions.

The point-slope form of a line equation is useful for writing the equation when you know the slope and a point on the line.
The point-slope form is given by: \[ y - y_1 = m(x - x_1) \], where \(m\) is the slope and \( (x_1, y_1) \) is a specific point on the line.
To use this form, simply substitute the known values for the slope and the point coordinates.
For instance, if a line has a slope \( m = -2 \) and passes through the point \( (4,1) \), you can form the equation as follows:
Substitute \( m \) with \( -2 \), \( x_1 \) with \( 4 \), and \( y_1 \) with \( 1 \) into the formula: \[ y - 1 = -2(x - 4) \].
This way, you directly get the equation of the line passing through the given point.

An equation of a line can be written in various forms, the most common being slope-intercept form and point-slope form.
The slope-intercept form is: \[ y = mx + b \], where \( m \) is the slope and \( b \) is the y-intercept (the point where the line crosses the y-axis).
In the point-slope form, as discussed earlier: \[ y - y_1 = m(x - x_1) \], the focus is on the slope and a specific point on the line.
Both forms are fundamental in representing linear relationships in algebra.
In the given exercise, the solution involved converting from point-slope form to matching the correct equation in slope-intercept form.
Understanding how to derive and manipulate these forms is crucial for solving linear equation problems effectively.