3x 2y 6 In Slope Intercept Form

Converting 3x + 2y = 6 to Slope-Intercept Form: A Comprehensive Guide

The equation 3x + 2y = 6 represents a straight line. While useful in its current form, converting it to slope-intercept form (y = mx + b) offers significant advantages, particularly when visualizing the line on a graph and understanding its characteristics. This comprehensive guide will walk you through the conversion process step-by-step, explaining the underlying concepts and providing practical examples. We’ll also explore how this form helps us understand the slope and y-intercept of the line, and delve into related concepts such as parallel and perpendicular lines.

Understanding Slope-Intercept Form (y = mx + b)

The slope-intercept form of a linear equation, y = mx + b, is a powerful tool for understanding and manipulating linear relationships. Let's break down what each component represents:

• y: The dependent variable, typically plotted on the vertical axis of a graph. It represents the output value.
• x: The independent variable, typically plotted on the horizontal axis of a graph. It represents the input value.
• m: The slope of the line. This value indicates the steepness and direction of the line. A positive slope indicates an upward trend (from left to right), while a negative slope indicates a downward trend. The slope is calculated as the change in y divided by the change in x (rise over run).
• b: The y-intercept. This is the point where the line intersects the y-axis (where x = 0). It represents the value of y when x is zero.

Converting 3x + 2y = 6 to Slope-Intercept Form

The goal is to manipulate the equation 3x + 2y = 6 to isolate y on one side of the equation, resulting in the form y = mx + b. Here's how we do it:

1. Subtract 3x from both sides:

   This step removes the x term from the left side, leaving only the y term:

   2y = -3x + 6

2. Divide both sides by 2:

   This isolates y, giving us the slope-intercept form:

   y = (-3/2)x + 3

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

Interpreting the Slope and Y-intercept

From the equation y = (-3/2)x + 3, we can directly extract the slope and y-intercept:

• Slope (m) = -3/2: This indicates a negative slope, meaning the line descends from left to right. The slope of -3/2 means that for every 2 units increase in x, y decreases by 3 units.

• Y-intercept (b) = 3: This means the line intersects the y-axis at the point (0, 3).

Graphing the Line

With the slope and y-intercept, graphing the line becomes straightforward:

1. Plot the y-intercept: Start by plotting the point (0, 3) on the y-axis.

2. Use the slope to find another point: The slope is -3/2. From the y-intercept (0, 3), move 2 units to the right (positive x-direction) and 3 units down (negative y-direction). This gives you a second point (2, 0).

3. Draw the line: Draw a straight line through the two points (0, 3) and (2, 0). This line represents the equation 3x + 2y = 6.

Parallel and Perpendicular Lines

Understanding the slope is crucial when working with parallel and perpendicular lines:

• Parallel Lines: Parallel lines have the same slope. Any line parallel to y = (-3/2)x + 3 will also have a slope of -3/2. For example, y = (-3/2)x + 5 is parallel to the original line because it shares the same slope but has a different y-intercept.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -3/2 is 2/3. Therefore, any line perpendicular to y = (-3/2)x + 3 will have a slope of 2/3. An example of a perpendicular line would be y = (2/3)x + 1.

Applications and Real-World Examples

The slope-intercept form has numerous applications across various fields:

• Economics: Modeling supply and demand curves. The slope represents the change in quantity demanded or supplied in response to a change in price.

• Physics: Representing velocity and displacement. The slope of a displacement-time graph gives the velocity.

• Engineering: Designing slopes for roads and ramps. The slope determines the steepness of the incline.

• Data Analysis: Visualizing trends and relationships in data. The slope and y-intercept can help identify patterns and make predictions.

Advanced Concepts and Extensions

Beyond the basic conversion and interpretation, several advanced concepts build upon this foundation:

• Finding the x-intercept: The x-intercept is the point where the line crosses the x-axis (where y = 0). To find it, set y = 0 in the equation y = (-3/2)x + 3 and solve for x. In this case, the x-intercept is 2.

• Systems of Linear Equations: Solving systems of linear equations involves finding the point where two or more lines intersect. The slope-intercept form simplifies this process by allowing direct comparison of slopes and y-intercepts.

• Linear Inequalities: Extending the concept to linear inequalities (e.g., y > (-3/2)x + 3) allows for representing regions on a graph rather than just a single line.

• Linear Transformations: Understanding how changes in the slope and y-intercept affect the line's position and orientation.

Conclusion

Converting the equation 3x + 2y = 6 to slope-intercept form (y = (-3/2)x + 3) provides a clear and concise representation of the line's characteristics. This form simplifies graphing, allows for easy determination of the slope and y-intercept, and facilitates the analysis of parallel and perpendicular lines. The fundamental concepts explored here form the basis for more advanced topics in linear algebra and have broad applications across numerous disciplines. Mastering this conversion is a key step in understanding and working with linear relationships. By understanding the slope, y-intercept, and their implications, you gain a powerful tool for analyzing and interpreting linear equations and their real-world applications. Remember that practice is key to solidifying your understanding, so work through various examples to build your confidence and expertise.