3x 2y 2 In Slope Intercept Form

Transforming 3x + 2y = 2 into Slope-Intercept Form: A Comprehensive Guide

The equation 3x + 2y = 2 represents a linear relationship between two variables, x and y. While useful in its current form, converting it to slope-intercept form (y = mx + b) offers significant advantages in understanding the line's characteristics, such as its slope (m) and y-intercept (b). This comprehensive guide will walk you through the process, explaining each step in detail and providing further insights into interpreting the resulting equation.

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

Before diving into the conversion, let's refresh our understanding of the slope-intercept form: y = mx + b.

y: Represents the dependent variable; its value depends on the value of x.
x: Represents the independent variable; its value is freely chosen.
m: Represents the slope of the line. The slope 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: Represents the y-intercept. This is the point where the line intersects the y-axis (where x = 0).

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

The goal is to isolate 'y' on one side of the equation. This involves a series of algebraic manipulations. Here's a step-by-step breakdown:

1. Subtract 3x from both sides:

The initial equation is: 3x + 2y = 2

Subtracting 3x from both sides gives: 2y = -3x + 2

2. Divide both sides by 2:

To isolate 'y', we divide both sides of the equation by 2:

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

3. Simplify:

Simplifying the equation gives us the slope-intercept form:

y = (-3/2)x + 1

Interpreting the Slope-Intercept Form: y = (-3/2)x + 1

Now that we've successfully converted the equation, let's analyze its components:

Slope (m = -3/2): The slope is -3/2. This negative slope indicates that the line slopes downwards from left to right. The value itself tells us that for every 2 units increase in x, y decreases by 3 units.
y-intercept (b = 1): The y-intercept is 1. This means the line crosses the y-axis at the point (0, 1).

Graphing the Line

With the slope-intercept form, graphing the line becomes straightforward. We can use the y-intercept as a starting point and then use the slope to find other points on the line.

Plot the y-intercept: Begin by plotting the point (0, 1) on the coordinate plane.
Use the slope to find another point: The slope is -3/2. This means we can move down 3 units and to the right 2 units from the y-intercept (0, 1) to find another point on the line. This gives us the point (2, -2).
Draw the line: Draw a straight line through the points (0, 1) and (2, -2). This line represents the equation y = (-3/2)x + 1.

Advantages of Slope-Intercept Form

Converting to slope-intercept form provides several advantages:

Easy Interpretation: The slope and y-intercept are directly visible, making it easy to understand the line's characteristics.
Simple Graphing: The y-intercept provides a starting point, and the slope allows for easy plotting of additional points.
Useful for Predictions: The equation allows for easy prediction of y values for any given x value, and vice versa.
Comparison with other lines: The slope-intercept form facilitates easy comparison of the slope and y-intercept with other lines, allowing for analysis of their relative positions and relationships.

Further Applications and Extensions

The slope-intercept form has numerous applications beyond simple graphing. It's fundamental in:

Linear Regression: In statistics, the slope-intercept form is crucial in representing linear regression models, which predict the relationship between variables.
Calculus: The slope is the derivative of the line, forming a basis for more advanced calculus concepts.
Computer Graphics: The slope-intercept form is used extensively in computer graphics for line drawing and other geometric calculations.
Physics and Engineering: Many physical and engineering phenomena can be modeled using linear equations, making the slope-intercept form a valuable tool.

Solving Problems using the Slope-Intercept Form

Let's illustrate the utility of the slope-intercept form with a few examples:

Example 1: Finding the y-value when x = 4:

Using the equation y = (-3/2)x + 1, we substitute x = 4:

y = (-3/2)(4) + 1 = -6 + 1 = -5

Therefore, when x = 4, y = -5.

Example 2: Finding the x-value when y = -1:

Substituting y = -1 into the equation:

-1 = (-3/2)x + 1

Solving for x:

-2 = (-3/2)x

x = (4/3)

Therefore, when y = -1, x = 4/3.

Conclusion

Transforming the equation 3x + 2y = 2 into its slope-intercept form, y = (-3/2)x + 1, provides invaluable insights into the line's properties. Understanding the slope and y-intercept enables straightforward graphing, facilitates predictions, and allows for comparisons with other linear equations. This conversion is a fundamental concept in algebra with wide-ranging applications across various fields. Mastering this process provides a solid foundation for more advanced mathematical concepts and real-world problem-solving. The simplicity and power of the slope-intercept form make it a crucial tool in the mathematician's and scientist's arsenal. Remember to practice these conversions to solidify your understanding and build confidence in tackling similar problems.