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The slope is a simple yet powerful concept in understanding how lines behave on a graph. Typically represented by the letter \( m \), the slope indicates the steepness and direction of a line. To calculate the slope between two points, use the formula:
\[m = \frac{{y_2-y_1}}{{x_2-x_1}}\]
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two designated points on the line. In simpler terms:

• Positive Slope: The line rises as it moves from left to right.
• Negative Slope: The line falls from left to right.
• Zero Slope: The line is horizontal.
• Undefined Slope: The line is vertical (division by zero occurs).

In our example, the slope of the line passing through the points \((-4,7)\) and \((2,4)\) is
\( m = -\frac{1}{2} \), indicating that the line gently falls as it moves right.

The point-slope form is a useful equation format to represent a line when you know one point on the line and its slope. The general form is represented as:
\[y - y_1 = m(x - x_1)\]
Here, \( (x_1, y_1) \) denotes a known point on the line, and \( m \) is the slope of the line. This equation makes it straightforward to plug in the values and solve for \( y \).
For example, using the point \((-4,7)\) and the slope \(-\frac{1}{2}\), the point-slope form becomes: \(y - 7 = -\frac{1}{2}(x + 4)\).
This form is incredibly handy for quickly forming line equations, especially when working directly from data points. Remember, the point-slope form centers around plugging in one point and utilizing the slope for computations.

The slope-intercept form is perhaps the most commonly used equation of a line, particularly because of its clarity in showing both the slope and the y-intercept. Written simply as:
\[y = mx + c\]
Where \( m \) is the slope and \( c \) is the y-intercept. This form tells you precisely where the line intersects the y-axis (i.e., at \( y = c \)).
From our exercise, after transforming the point-slope form equation, we arrived at the slope-intercept form: \( y = -\frac{1}{2}x + 9 \).
Here, \( -\frac{1}{2} \) serves as the slope, and \( 9 \) is the y-intercept, indicating that our line crosses the y-axis at 9. The slope-intercept form is favored for its simplicity and ease in plotting lines quickly on a graph.