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The slope in a linear equation is a measure that tells you how steep the line is. It indicates the direction and angle of the line. In our equation, \[ y = \frac{1}{3}x - \frac{3}{2} \]we find that the slope \( m \) is \( \frac{1}{3} \). To understand this, think of the slope as a ratio of the change in the \( y \)-axis (also known as rise) compared to the change in the \( x \)-axis (run). Here is what this means:

• \( \frac{1}{3} \) means for every 1 unit of vertical change, the line moves 3 units horizontally.
• This allows us to predict where another point on the line will be, given one known point.

The slope can tell us whether a line will ascend or descend as we move from left to right across the graph. Since our slope is positive, the line rises as it moves from left to right.

The \( y \)-intercept is the point where the line crosses the \( y \)-axis. In the slope-intercept form, \[ y = mx + b \]\( b \) represents the \( y \)-intercept. In our equation, \[ y = \frac{1}{3}x - \frac{3}{2} \]\( b = -\frac{3}{2} \). Thus, the line crosses the \( y \)-axis at the point (0, -\frac{3}{2}). To visualize this:

• The \( y \)-intercept is the value of \( y \) when \( x \) is zero.
• This specific point helps to anchor the line as it gives an exact location on the \( y \)-axis where the line begins or intersects.

Understanding the \( y \)-intercept is crucial for accurately sketching the graph. It is often the starting point when drawing the line.

Graphing linear equations involves multiple steps, integrating both the slope and the \( y \)-intercept into the visualization on a coordinate plane. Let's break down how we can graph our specific equation:1. **Locate the Y-Intercept**: Begin by plotting the \( y \)-intercept, where our line crosses the \( y \)-axis, which is point (0, -\frac{3}{2}). This point is crucial as it serves as a starting reference.
2. **Use Slope to Find Another Point**: From the \( y \)-intercept, use the slope \( \frac{1}{3} \) to find another point on the graph. For instance, move 3 units to the right (positive x-direction) and 1 unit up (positive y-direction) to find the next point, (3, -\frac{1}{2}).
3. **Draw the Line**: With at least two points plotted, draw a straight line through them extending in both directions. This represents the infinite set of solutions the equation provides.
These steps ensure that the graph accurately represents the equation, allowing visual understanding and verification of the line's behavior.