Graphing the Linear Equation 2x - 3y 0: A Step-by-Step Guide

Understanding how to graph a linear equation is a fundamental skill in algebra. In this guide, we'll demonstrate the process of graphing the equation 2x - 3y 0. This involves multiple steps, from rearranging the equation to identify its slope and y-intercept, to plotting the points and drawing the line.

Step 1: Rearrange the Equation

The first step in graphing 2x - 3y 0 is to rearrange the equation to express y in terms of x. This is done in the slope-intercept form y mx b, where m is the slope and b is the y-intercept.

Starting with 2x - 3y 0, we first isolate y:

2x - 3y 0

3y 2x

y 2/3x

Step 2: Identify the Slope and Y-Intercept

The equation y 2/3x is now in the slope-intercept form. Here,

m 2/3 (the slope)

b 0 (the y-intercept)

Step 3: Plot the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. Since b 0, the y-intercept is at the origin, (0, 0).

Step 4: Use the Slope to Find Another Point

The slope, 2/3, tells us how much y changes for a given change in x. This can be thought of as rise over run, where the rise is 2 units upward and the run is 3 units to the right.

Starting from the y-intercept (0, 0), move up 2 units and right 3 units to find another point on the line:

Rise: 2 units up from (0, 0) gives (0, 2) Run: 3 units to the right from (0, 2) gives (3, 2)

Step 5: Draw the Line

Plot the points (0, 0) and (3, 2) on a graph. Then, draw a straight line through these points. This line represents all solutions to the equation 2x - 3y 0.

Graph Visualization

A rough visualization of the graph:

In this graph, the line passes through the origin (0, 0) with a slope of 2/3. The line extends in both directions to represent all the solutions to the equation.

Additional Points (Optional)

To ensure clarity, instructors might ask for more points. Additional points, such as (0, 0), (3, 2), and (-3, -2), can be plotted to confirm the line:

x -3 and y (-2) gives the point (-3, -2), since y 2/3x

Plot these points and connect them with a straight line to visualize the equation graphically.

Conclusion

Graphing the linear equation 2x - 3y 0 involves identifying its slope and y-intercept, plotting key points, and drawing a line through them. This step-by-step process helps in understanding and visualizing linear equations in a more concrete manner.

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