Exploring the Graph Equation x - 3y 6: A Comprehensive Guide

The equation x - 3y 6 is a linear equation in two unknowns, x and y. It represents a straight line when graphed on an xOy coordinate plane. In this guide, we will delve into the details of this equation, how to find its intercepts, and how to graph it. We'll also discuss the significance of the slope and intercepts in interpreting this linear equation.

Understanding the Equation and Graphing it

The equation x - 3y 6 can be transformed to the slope-intercept form y mx c. Let's explore this transformation step-by-step:

Start with the equation: x - 3y 6 Rearrange the equation to isolate y: -3y -x 6 Divide both sides by -3 to solve for y: y 1/3x - 2

This transformation reveals that the equation is a linear function with a slope of 1/3 (or tan θ where θ is the angle the line makes with the x-axis) and a y-intercept of -2.

To graph this equation, we can identify two key points:

Finding the Intercepts

x-intercept (point where y 0):
Set y 0 in the equation x - 3y 6.
x - 0 6 -> x 6
The x-intercept is (6, 0).

y-intercept (point where x 0):
Set x 0 in the equation x - 3y 6.
0 - 3y 6 -> y -2
The y-intercept is (0, -2).

Plotting these two points (6, 0) and (0, -2) and drawing a straight line through them will graph the equation x - 3y 6.

Interpreting the Graph

The graph of x - 3y 6 represents a straight line with a positive slope of 1/3. This means that for every unit increase in x, y increases by 1/3. The y-intercept of -2 indicates where the line crosses the y-axis. The x-intercept at 6 means the line crosses the x-axis at 6.

Real-World Application

Linear equations like x - 3y 6 have numerous real-world applications. For instance, this equation could represent a relationship between two variables in economics, physics, engineering, or any field where a linear relationship is present. By understanding the slope and intercepts, we can make predictions or solve for unknown values.

Example: If the equation represented a cost-benefit relationship, the slope could indicate the rate of change in cost per unit benefit. The x-intercept would indicate the point at which there is no benefit, and the y-intercept would show the initial cost before any benefits are realized.

Conclusion

Understanding the graph equation x - 3y 6 involves recognizing its linear form, transforming it to the slope-intercept form, finding its intercepts, and interpreting the significance of these points. This equation represents a fundamental concept in algebra and has wide-ranging applications in various fields.