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The slope-intercept form is a specific way to express the equation of a straight line. It is one of the most common forms for linear equations due to its simplicity and ease of understanding. The general formula for the slope-intercept form is \( y = mx + b \). Here’s a breakdown of what each component means:

• \( y \): Represents the output or the dependent variable in the equation.
• \( m \): Denotes the slope of the line, indicating how steep the line is.
• \( x \): Represents the input or independent variable.
• \( b \): Is the \( y \)-intercept, the point where the line crosses the y-axis.

By substituting the slope \( m \) and \( y \)-intercept \( b \) into the equation, it provides a complete description of the line. This form makes it very easy to graph a line on a coordinate plane, as you can start at the \( y \)-intercept \( b \) on the y-axis, then use the slope \( m \) to determine the rise over run for additional points on the line.

The slope of a line, represented by \( m \), is a measure of its steepness and direction. It shows the rate at which the line rises or falls as you move along it. Slope is an essential component of linear equations and is defined mathematically as the change in \( y \) over the change in \( x \), or "rise over run."

Mathematically, the slope \( m \) can be calculated using the formula:
\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\]

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A zero slope indicates a horizontal line, which is flat.
• An undefined slope denotes a vertical line, where there is no horizontal change.

In our example, the given slope is \( -3 \), indicating that for every 1 unit we move to the right (along the x-axis), the line drops down 3 units (along the y-axis). Understanding the slope helps in predicting and analyzing how variables are related.

The \( y \)-intercept of a line, designated as \( b \) in the slope-intercept form, is the point where the line crosses the y-axis. This is the value of \( y \) when \( x \) is equal to zero. It is a pivotal point that gives a starting place for graphing the line based on its equation.

To find the \( y \)-intercept from an equation:

• Set \( x = 0 \).
• Solve for \( y \) to find the intercept value.

In mathematical terms, if you substitute \( x = 0 \) into the equation \( y = mx + b \), you get \( y = b \).

For our exercise, the \( y \)-intercept is \( 9 \). This allows us to start graphing the line by placing a point at \( (0, 9) \) on the coordinate plane. From this point, the slope will dictate the line's angle and direction. The \( y \)-intercept is crucial for determining the initial position of the line in relation to the y-axis.