3x 2y 4 In Slope Intercept Form

Converting 3x + 2y = 4 to Slope-Intercept Form: A Comprehensive Guide

The equation 3x + 2y = 4 represents a straight line. However, it's not in the easily interpretable slope-intercept form, which is y = mx + b. In this form, 'm' represents the slope of the line (how steep it is), and 'b' represents the y-intercept (where the line crosses the y-axis). This guide will walk you through the process of converting 3x + 2y = 4 into slope-intercept form, explaining each step in detail and exploring the properties of the resulting equation.

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

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

• y: Represents the y-coordinate of any point on the line.
• x: Represents the x-coordinate of any point on the line.
• m: Represents the slope of the line. The slope describes 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. A slope of zero represents a horizontal line. The slope is calculated as the change in y divided by the change in x (rise over run).
• b: Represents the y-intercept, which is the point where the line intersects the y-axis (where x = 0).

Converting 3x + 2y = 4 to Slope-Intercept Form

Our goal is to isolate 'y' on one side of the equation to obtain the form y = mx + b. Let's follow these steps:

1. Subtract 3x from both sides: This step aims to move the term containing 'x' to the right side of the equation.

   3x + 2y - 3x = 4 - 3x

   This simplifies to:

   2y = -3x + 4

2. Divide both sides by 2: This step isolates 'y', giving us the desired slope-intercept form.

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

   This simplifies to:

   y = (-3/2)x + 2

Now we have successfully converted the equation 3x + 2y = 4 into slope-intercept form: y = (-3/2)x + 2

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

Having obtained the slope-intercept form, let's analyze its components:

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

• Y-intercept (b) = 2: This indicates that the line intersects the y-axis at the point (0, 2). This is the point where x = 0.

Graphing the Equation: y = (-3/2)x + 2

The slope-intercept form makes graphing the equation extremely straightforward. We can use the y-intercept as a starting point and then use the slope to find other points on the line.

1. Plot the y-intercept: Start by plotting the point (0, 2) on a Cartesian coordinate system.

2. Use the slope to find another point: The slope is -3/2. This means that from the y-intercept (0,2), we can move 2 units to the right and 3 units down to find another point on the line. This brings us to the point (2, -1).

3. Draw the line: Draw a straight line passing through the points (0, 2) and (2, -1). This line represents the equation y = (-3/2)x + 2.

Finding the x-intercept

While the y-intercept is readily available from the slope-intercept form, we can also find the x-intercept, which is the point where the line intersects the x-axis (where y = 0). To find this, substitute y = 0 into the original equation or the slope-intercept form:

Using the slope-intercept form:

0 = (-3/2)x + 2

Solving for x:

(3/2)x = 2

x = (2 * 2) / 3 = 4/3

Therefore, the x-intercept is (4/3, 0).

Alternative Methods for Conversion

While the method detailed above is the most straightforward, there are other ways to convert the equation to slope-intercept form. For example, you could use the following techniques:

• Solving for y using algebraic manipulation: This involves strategically applying addition, subtraction, multiplication, and division to isolate y. This method directly follows the steps shown above.

• Using matrices: This method is more suitable for more complex systems of equations, but it's applicable here. This method would involve writing the given equation in matrix form and applying row reduction techniques to solve for y. This is generally not the most efficient method for this particular example but is worth noting for broader applications in linear algebra.

Applications and Significance of the Slope-Intercept Form

The slope-intercept form is crucial in various fields, including:

• Mathematics: It's fundamental in algebra, calculus, and linear algebra for understanding and manipulating linear equations.

• Physics: Used extensively to model linear relationships between physical quantities, such as velocity and time, or force and displacement.

• Economics: Helpful in representing linear relationships between economic variables like supply and demand.

• Engineering: Used for modeling and analyzing various systems.

• Computer graphics: Essential for defining lines and other geometric shapes in 2D and 3D graphics.

• Data analysis: Used to create trend lines and make predictions based on linear relationships within datasets.

Real-World Examples

Consider the following real-world applications where the slope-intercept form proves valuable:

• Calculating the cost of a taxi ride: A taxi might charge a fixed initial fee (y-intercept) plus a charge per mile (slope). The slope-intercept form could accurately model the total cost based on the distance traveled.

• Predicting plant growth: The growth of a plant over time can often be approximated using a linear model. The slope represents the growth rate, while the y-intercept represents the initial height.

Conclusion

Converting the equation 3x + 2y = 4 to slope-intercept form, y = (-3/2)x + 2, allows for a clear understanding of the line's properties, namely its slope and y-intercept. This form simplifies graphing, calculating intercepts, and applying the equation in various mathematical and real-world contexts. Understanding these concepts is crucial for success in various academic and professional pursuits. The steps involved in this conversion are fundamental to solving linear equations and interpreting their graphical representations. Mastering this skill lays the foundation for tackling more complex mathematical problems and enhances problem-solving abilities.