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The slope-intercept form is a way of writing the equation of a line. It is expressed as \( y = mx + b \).
This format is particularly useful because it clearly shows two key features of the line:

• Slope \( (m) \): This represents the steepness or tilt of the line. It tells us how much the line rises or falls for each step you take to the right along the x-axis.
• Y-Intercept \( (b) \): This is the point where the line crosses the y-axis. It represents the value of \( y \) when \( x = 0 \).

In our exercise, we start with the equation \(-\frac{1}{2} x + 2y = 9\). To convert it into slope-intercept form, solve for \( y \): First, rearrange the equation: \( 2y = \frac{1}{2} x + 9 \). Next, divide each term by 2: \( y = \frac{1}{4} x + \frac{9}{2} \). This newly arranged equation allows us to identify the slope \( m = \frac{1}{4} \) and the y-intercept \( b = \frac{9}{2} \).

Graphing equations involves plotting points and drawing a line through these points on a coordinate plane. Knowing the slope-intercept form of a line \( y = mx + b \) makes this task easier, as it provides a starting point and direction.
Begin by plotting the y-intercept. This is the point where the line crosses the y-axis. For our equation \( y = \frac{1}{4} x + \frac{9}{2} \), the y-intercept is \( \frac{9}{2} \), or 4.5 when expressed in decimal form.

• Plot the point (0, 4.5) on the y-axis. This is your starting point.

Next, use the slope to determine the direction of the line. The slope \( \frac{1}{4} \) means that for every 4 units you move to the right along the x-axis, move 1 unit up.

• From (0, 4.5), move right to (4, 5.5).
• Plot this second point (4, 5.5) on the graph.

Finally, connect the points with a straight line using a ruler. This line is the visual representation of the equation.

The concept of slope is fundamental in understanding linear equations. Slope ( \( m \)) is calculated as the "rise over run," which is the change in \( y \) divided by the change in \( x \).
The formula for calculating slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula subtracts the y-values of two points and divides it by the x-values' difference.
In our exercise, the slope \( m = \frac{1}{4} \) was directly obtained from the slope-intercept form \( y = \frac{1}{4} x + \frac{9}{2} \). This means that:

• For every 4 units you move horizontally to the right on the graph (increasing \( x \)), the line will rise 1 unit vertically (increasing \( y \)).

If we were given points such as (0, 4.5) and (4, 5.5), we could also find the slope by plugging into the slope formula:

• \( m = \frac{5.5 - 4.5}{4 - 0} = \frac{1}{4} \).

Thus, understanding the slope helps you understand how steep the line is and how it visually appears on a graph.