3x 2y 8 In Slope Intercept Form

Converting 3x + 2y = 8 to Slope-Intercept Form: A Comprehensive Guide

The equation 3x + 2y = 8 represents a straight line. However, it's not in the widely used slope-intercept form, which is y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. This guide will walk you through the process of converting the given equation into slope-intercept form, explaining each step thoroughly and exploring related concepts. We'll also delve into the practical applications and implications of this transformation.

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

Before we begin the conversion, let's solidify our understanding of the slope-intercept form. This form provides a concise and insightful representation of a linear equation:

m (slope): This value indicates the steepness and direction of the line. A positive slope signifies 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 (y-intercept): This value represents the point where the line intersects the y-axis. In other words, it's the y-coordinate when x = 0.

The slope-intercept form allows us to quickly identify key characteristics of the line and easily plot it on a graph.

Converting 3x + 2y = 8 to Slope-Intercept Form

Now, let's convert the equation 3x + 2y = 8 into the slope-intercept form (y = mx + b). We need to isolate 'y' on one side of the equation. Here's a step-by-step breakdown:

Subtract 3x from both sides:

This step aims to move the 'x' term to the right side of the equation. Subtracting 3x from both sides, we get:

2y = -3x + 8
Divide both sides by 2:

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

y = (-3/2)x + 4

Therefore, the equation 3x + 2y = 8 in slope-intercept form is y = (-3/2)x + 4.

Analyzing the Result: Slope and Y-intercept

Now that we've converted the equation, let's analyze the resulting slope-intercept form:

Slope (m) = -3/2: This indicates a negative slope, meaning the line slopes downwards from left to right. The magnitude of the slope (3/2) tells us that for every 2 units of movement along the x-axis, the line moves 3 units down along the y-axis.
Y-intercept (b) = 4: This tells us that the line intersects the y-axis at the point (0, 4).

Graphical Representation

Plotting the line on a graph is straightforward using the slope-intercept form. We know the y-intercept is 4, so we can plot the point (0, 4). From this point, we can use the slope (-3/2) to find another point on the line. Since the slope is -3/2, we move 2 units to the right and 3 units down, leading to the point (2, 1). By connecting these two points, we obtain the graph of the line.

Practical Applications and Importance

Converting an equation to slope-intercept form has numerous applications across various fields:

Economics: In economics, linear equations are often used to model relationships between variables such as supply and demand. The slope-intercept form provides a clear representation of these relationships, allowing economists to analyze the impact of changes in one variable on another.
Physics: In physics, linear equations are used to describe motion and other physical phenomena. The slope-intercept form is essential for understanding the velocity and displacement of objects.
Engineering: Engineers utilize linear equations to model various systems and processes. The slope-intercept form is crucial for analyzing the behavior of these systems and optimizing their performance.
Data Analysis: In data analysis, linear regression is a common technique used to model the relationship between variables. The slope-intercept form provides a readily interpretable representation of the regression line, allowing analysts to understand the relationship between the variables.
Computer Graphics: In computer graphics, linear equations are used to represent lines and other geometric shapes. The slope-intercept form is essential for rendering these shapes accurately on a computer screen.

Alternative Forms and Conversions

While the slope-intercept form is widely used, it's not the only way to represent a linear equation. Other common forms include:

Standard Form (Ax + By = C): This form is useful for certain calculations and is easily convertible to slope-intercept form.
Point-Slope Form (y - y1 = m(x - x1)): This form is particularly useful when you know the slope and one point on the line.

Understanding these different forms and the ability to convert between them enhances problem-solving capabilities in various mathematical and real-world applications.

Solving Problems Using the Slope-Intercept Form

Let's consider a couple of example problems to demonstrate the utility of the slope-intercept form:

Problem 1: Find the y-coordinate when x = 6 in the equation y = (-3/2)x + 4.

Solution: Substitute x = 6 into the equation:

y = (-3/2)(6) + 4 = -9 + 4 = -5

Therefore, the y-coordinate is -5 when x = 6.

Problem 2: Determine if the point (4, -2) lies on the line represented by y = (-3/2)x + 4.

Solution: Substitute x = 4 and y = -2 into the equation:

-2 = (-3/2)(4) + 4

-2 = -6 + 4

-2 = -2

Since the equation holds true, the point (4, -2) lies on the line.

Conclusion

Converting the equation 3x + 2y = 8 to its slope-intercept form, y = (-3/2)x + 4, provides a concise and insightful representation of the line. This form reveals the slope (-3/2) and the y-intercept (4), enabling easy graphing and analysis. Understanding this conversion is fundamental in various fields, from economics and physics to data analysis and computer graphics. Mastering this skill significantly enhances problem-solving capabilities and provides a deeper understanding of linear relationships. The ability to seamlessly transition between different equation forms further broadens mathematical comprehension and facilitates practical applications across diverse disciplines. The examples provided showcase the practical utility of the slope-intercept form in solving real-world problems. Therefore, a thorough understanding of this form is crucial for success in many areas of study and professional endeavors.