graph 2y 3x 6

2 min read 20-10-2024

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

This article will guide you through understanding and graphing the linear equation 2y + 3x = 6. We'll explore key concepts like slope-intercept form and intercepts, and ultimately visualize the equation as a straight line.

1. What is the Equation Telling Us?

The equation 2y + 3x = 6 represents a linear relationship between two variables, x and y. This means that the points satisfying the equation form a straight line when plotted on a coordinate plane.

2. Converting to Slope-Intercept Form (y = mx + c)

The most common form for understanding a linear equation is the slope-intercept form: y = mx + c. Here, 'm' represents the slope of the line, and 'c' is the y-intercept (where the line crosses the y-axis).

To convert our equation into slope-intercept form, we need to solve for y:

Subtract 3x from both sides: 2y = -3x + 6
Divide both sides by 2: y = (-3/2)x + 3

Now we have our equation in the slope-intercept form: y = (-3/2)x + 3

3. Interpreting the Slope and Y-intercept

Slope (m = -3/2): This tells us the line's steepness and direction. A negative slope indicates a downward slant from left to right. The value -3/2 means that for every 2 units we move to the right, we move 3 units down.
Y-intercept (c = 3): This tells us where the line crosses the y-axis. In this case, the line crosses the y-axis at the point (0, 3).

4. Graphing the Line

Now that we have the slope and y-intercept, we can easily graph the line:

Plot the y-intercept: Mark the point (0, 3) on the y-axis.
Use the slope to find another point: From the y-intercept, move 2 units to the right and 3 units down. This brings us to the point (2, 0).
Draw the line: Connect the two points with a straight line.

You have now successfully graphed the equation 2y + 3x = 6!

5. Additional Insights

X-intercept: To find where the line crosses the x-axis, we can set y = 0 in the original equation. Solving for x, we get x = 2. Therefore, the x-intercept is (2, 0).
Using a Table of Values: Alternatively, you can create a table of values by choosing different values of x and solving for y. This will give you multiple points that you can plot to draw the line.

Conclusion

By understanding slope-intercept form and utilizing its key components, we were able to efficiently graph the equation 2y + 3x = 6. This knowledge is crucial for understanding linear relationships and visualizing them on a coordinate plane.