91Ó°ÊÓ

The slope of a line is a number that describes its steepness and direction. In the slope-intercept form equation, which is written as \( y = mx + b \), the slope is represented by the symbol \( m \). Simply put, the slope tells us how many units the line goes up or down for each unit it moves to the right.

For example, in the equation \( y = \frac{3}{2}x - 6 \), the slope is \( \frac{3}{2} \). This means the line rises 3 units upward for every 2 units it goes to the right. It's a positive slope, so the line will go upwards from left to right.

Understanding the slope:

• If the slope is positive, the line goes upward as you move from left to right.
• If the slope is negative, the line goes downward.
• If the slope is zero, the line is horizontal.
• If the slope is undefined, the line is vertical.

The slope helps us quickly grasp the angle and orientation of the line, which is crucial when graphing linear equations.

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept equation \( y = mx + b \), the y-intercept is represented by the symbol \( b \).

In more straightforward terms, the y-intercept is the value of \( y \) when \( x = 0 \). In the line equation \( y = \frac{3}{2}x - 6 \), the y-intercept is \(-6\). This means that when the line crosses the y-axis, it does so at the point \((0, -6)\).

Knowing the y-intercept is helpful for:

• Finding a starting point on a graph.
• Understanding the initial value of a relationship.
• Quickly identifying how a line is positioned on the graph.

The y-intercept gives us an easy and quick reference for plotting the line on a graph.

Graphing linear equations involves drawing a straight line that represents all solutions of the equation. To graph a line using the slope-intercept form \( y = mx + b \), follow these steps:

1. **Plot the y-intercept:** Find the intersection of the line with the y-axis. In our example, the y-intercept is \(-6\), so you start by plotting a point at \((0, -6)\).

2. **Use the slope to find another point:** The slope tells you how to find the next point by moving vertically and horizontally. With a slope of \(\frac{3}{2}\), move 3 units up and 2 units to the right from your starting point at \((0, -6)\). This brings you to point \((2, -3)\).

3. **Draw the line:** Connect the points you plotted with a straight line extending in both directions.

Graphing linear equations is a powerful visualization tool. It helps us see the relationship between variables and make predictions based on the graph.