3y 2x 6 graph

Guri Bee

2 min read

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20-10-2024

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Understanding the Graph of 3y = 2x + 6: A Step-by-Step Guide

The equation 3y = 2x + 6 represents a straight line. This article will guide you through understanding how to graph this equation and explore its key features.

1. Finding the Slope and y-intercept

The equation is currently in standard form (Ax + By = C). To easily identify the slope and y-intercept, we'll convert it to slope-intercept form (y = mx + c), where 'm' is the slope and 'c' is the y-intercept.

Step 1: Isolate 'y'

Divide both sides of the equation by 3:

3y/3 = (2x + 6)/3

Step 2: Simplify the equation

y = (2/3)x + 2

Now, the equation is in slope-intercept form.

Key Observations:

• Slope (m): The coefficient of 'x' is 2/3. This means for every 3 units you move to the right on the x-axis, you move 2 units up on the y-axis.
• y-intercept (c): The constant term is 2. This tells us the line crosses the y-axis at the point (0, 2).

2. Plotting the Graph

Step 1: Plot the y-intercept

Mark the point (0, 2) on the graph.

Step 2: Use the slope to find another point

Since the slope is 2/3, move 3 units to the right from the y-intercept and then 2 units up. This brings you to the point (3, 4). Plot this point.

Step 3: Draw the line

Draw a straight line that passes through the two points you plotted. This line represents the graph of the equation 3y = 2x + 6.

3. Key Features and Interpretations

• Positive Slope: The positive slope (2/3) indicates that the line is increasing as you move from left to right. This means as the value of 'x' increases, the value of 'y' also increases.
• y-intercept: The y-intercept (0, 2) tells us the point where the line intersects the y-axis.
• Linear Relationship: The graph of 3y = 2x + 6 represents a linear relationship. This means that there is a constant rate of change between the variables 'x' and 'y'.

Additional Notes and Examples

1. Practical Application:

This equation could represent a relationship between the number of hours worked (x) and the amount earned (y). The slope (2/3) would mean that for every 3 hours worked, the person earns $2. The y-intercept (2) could represent a fixed hourly rate or a bonus.

2. Finding x-intercept:

To find the x-intercept, we need to find the point where the line crosses the x-axis. This happens when y = 0. Substituting y = 0 in the equation, we get:

0 = (2/3)x + 2

Solving for 'x', we get x = -3. Therefore, the x-intercept is (-3, 0).

3. Using Technology:

Many online tools and graphing calculators can help you visualize the graph of equations. Just input the equation 3y = 2x + 6 and the tool will generate the graph for you. This can be helpful for checking your work and exploring different scenarios.

Remember: Understanding the relationship between equations and their graphs is crucial in many areas of mathematics, science, and engineering. By practicing and applying the concepts discussed in this article, you can gain a solid understanding of linear functions and their applications.