3x 2y 10 In Slope Intercept Form

Converting 3x + 2y = 10 to Slope-Intercept Form: A Comprehensive Guide

The equation 3x + 2y = 10 represents a straight line. However, it's not in the slope-intercept form, which is widely preferred for its clarity and ease of use in various applications. This comprehensive guide will walk you through the steps of converting this equation into slope-intercept form (y = mx + b), explaining the concepts involved and providing practical examples. We'll also explore the significance of the slope (m) and the y-intercept (b), and delve into how this form facilitates graphing and problem-solving.

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

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

• y: Represents the dependent variable, typically plotted on the vertical axis of a graph.
• x: Represents the independent variable, typically plotted on the horizontal axis of a graph.
• 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. 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 crosses the y-axis (where x = 0).

Converting 3x + 2y = 10 to Slope-Intercept Form

The key to converting 3x + 2y = 10 into slope-intercept form is to isolate 'y' on one side of the equation. We'll achieve this through a series of algebraic manipulations:

1. Subtract 3x from both sides:

   This step removes the '3x' term from the left side, leaving only the '2y' term:

   2y = -3x + 10

2. Divide both sides by 2:

   This isolates 'y' and gives us the slope-intercept form:

   y = (-3/2)x + 5

Therefore, the slope-intercept form of the equation 3x + 2y = 10 is y = (-3/2)x + 5.

Interpreting the Slope and Y-Intercept

Now that we have the equation in slope-intercept form, let's analyze the slope and y-intercept:

• Slope (m = -3/2): The slope is -3/2. This negative slope indicates that the line slopes downward 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 = 5): The y-intercept is 5. This means the line intersects the y-axis at the point (0, 5).

Graphing the Equation

With the slope and y-intercept, graphing the equation is straightforward:

1. Plot the y-intercept: Begin by plotting the point (0, 5) on the y-axis.

2. Use the slope to find another point: Since the slope is -3/2, move 2 units to the right (positive x-direction) and 3 units down (negative y-direction) from the y-intercept. This gives us the point (2, 2).

3. Draw the line: Draw a straight line passing through the points (0, 5) and (2, 2). This line represents the equation 3x + 2y = 10.

Practical Applications and Problem Solving

The slope-intercept form is invaluable in various applications, including:

• Predicting values: Given an x-value, you can easily calculate the corresponding y-value using the equation y = (-3/2)x + 5. For instance, if x = 4, y = (-3/2)(4) + 5 = -1.

• Finding intercepts: The y-intercept is readily available (5). To find the x-intercept (where y = 0), set y = 0 in the equation and solve for x: 0 = (-3/2)x + 5; x = 10/3. This gives us the x-intercept (10/3, 0).

• Comparing lines: The slope-intercept form facilitates comparing the slopes and y-intercepts of different lines, determining if they are parallel (same slope, different y-intercept), perpendicular (slopes are negative reciprocals), or neither.

• Modeling real-world situations: Linear equations are often used to model real-world relationships between variables. The slope-intercept form provides a clear and concise way to represent these relationships. For example, it could model the relationship between the number of hours worked (x) and the total earnings (y).

Advanced Considerations and Related Concepts

Let's delve into some more advanced topics related to linear equations and the slope-intercept form:

1. Parallel and Perpendicular Lines

• Parallel Lines: Two lines are parallel if they have the same slope (m) but different y-intercepts (b). For instance, y = (-3/2)x + 5 and y = (-3/2)x + 10 are parallel lines.

• Perpendicular Lines: Two lines are perpendicular if their slopes are negative reciprocals of each other. The negative reciprocal of -3/2 is 2/3. Therefore, y = (-3/2)x + 5 is perpendicular to any line with a slope of 2/3, such as y = (2/3)x + 7.

2. Standard Form and Point-Slope Form

While the slope-intercept form is widely used, other forms of linear equations exist:

• Standard Form (Ax + By = C): This form, where A, B, and C are constants, is useful for certain applications but doesn't directly reveal the slope and y-intercept. The equation 3x + 2y = 10 is in standard form.

• Point-Slope Form (y - y1 = m(x - x1)): This form is particularly useful when you know the slope (m) and a point (x1, y1) on the line.

3. Applications in Calculus and Beyond

The foundation of understanding linear equations and their different forms, like the slope-intercept form, extends into more advanced mathematical concepts. In calculus, for example, the slope represents the instantaneous rate of change of a function at a specific point. The concepts of slope and intercepts are fundamental in numerous fields, including physics, engineering, economics, and computer science.

Conclusion: Mastering the Slope-Intercept Form

Converting the equation 3x + 2y = 10 to its slope-intercept form, y = (-3/2)x + 5, provides a more intuitive and readily applicable representation of the line. This form simplifies graphing, problem-solving, and comparison with other lines. Understanding the significance of the slope and y-intercept opens doors to advanced concepts and applications across various disciplines. By mastering the slope-intercept form, you equip yourself with a powerful tool for understanding and working with linear equations.