4x 3y 12 In Slope Intercept Form

4x + 3y = 12 in Slope-Intercept Form: A Comprehensive Guide

Converting the equation 4x + 3y = 12 into slope-intercept form (y = mx + b) is a fundamental concept in algebra. This form reveals crucial information about the line's characteristics: its slope (m) and its y-intercept (b). This comprehensive guide will walk you through the process step-by-step, explore related concepts, and offer practical applications.

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

The slope-intercept form, y = mx + b, is a powerful tool for representing linear equations. Let's break down its components:

• y: Represents the dependent variable, typically plotted on the vertical axis.
• x: Represents the independent variable, typically plotted on the horizontal axis.
• 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. A slope of zero indicates a horizontal line. An undefined slope indicates a vertical 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 crosses the y-axis (where x = 0).

Converting 4x + 3y = 12 to Slope-Intercept Form

Our goal is to transform the equation 4x + 3y = 12 into the form y = mx + b. This involves isolating 'y' on one side of the equation. Here's how:

1. Subtract 4x from both sides:

   This step aims to move the 'x' term to the right side of the equation. This gives us:

   3y = -4x + 12

2. Divide both sides by 3:

   To isolate 'y', we need to divide every term by the coefficient of 'y', which is 3:

   y = (-4/3)x + 4

Now we have the equation in slope-intercept form: y = (-4/3)x + 4

Interpreting the Results

From the slope-intercept form, y = (-4/3)x + 4, we can extract the following information:

• Slope (m) = -4/3: This indicates a negative slope, meaning the line will slant downwards from left to right. The slope of -4/3 means that for every 3 units of increase in x, y decreases by 4 units.

• Y-intercept (b) = 4: This means the line crosses the y-axis at the point (0, 4).

Graphing the Line

With the slope and y-intercept, we can easily graph the line.

1. Plot the y-intercept: Start by plotting the point (0, 4) on the y-axis.

2. Use the slope to find another point: The slope is -4/3. This can be interpreted as a rise of -4 and a run of 3. From the y-intercept (0, 4), move down 4 units and to the right 3 units. This will give you another point on the line: (3, 0).

3. Draw the line: Connect the two points (0, 4) and (3, 0) with a straight line. This line represents the equation 4x + 3y = 12.

Applications of Slope-Intercept Form

The slope-intercept form is widely used across various fields. Here are some examples:

• Economics: Representing supply and demand curves. The slope of the demand curve shows the relationship between price and quantity demanded.

• Physics: Describing motion. The slope of a velocity-time graph represents acceleration.

• Engineering: Modeling linear relationships between variables in various systems.

• Finance: Analyzing trends in stock prices or other financial data. A linear regression might be used to fit a line to the data, providing a slope and y-intercept to forecast future values.

• Data Science: Linear regression is a fundamental statistical tool used extensively to model linear relationships in data and make predictions. The slope and intercept from the regression line provide insights into the relationship between variables.

Parallel and Perpendicular Lines

Understanding slope is crucial for determining the relationship between different lines.

• Parallel Lines: Parallel lines have the same slope. If you have a line in slope-intercept form and want to find a parallel line, it will share the same 'm' value. For example, any line of the form y = (-4/3)x + c, where 'c' is any constant, would be parallel to y = (-4/3)x + 4.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. If a line has a slope of 'm', a line perpendicular to it will have a slope of -1/m. For our line with a slope of -4/3, a perpendicular line would have a slope of 3/4.

Solving Problems Using Slope-Intercept Form

Let's consider a few example problems:

Problem 1: Find the equation of a line parallel to y = (-4/3)x + 4 and passing through the point (3, 2).

Solution: Since the line is parallel, it will have the same slope, -4/3. Using the point-slope form (y - y1 = m(x - x1)), we have:

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

Simplifying this equation gives us:

y = (-4/3)x + 6

Problem 2: Find the equation of a line perpendicular to y = (-4/3)x + 4 and passing through the point (6, 1).

Solution: A perpendicular line will have a slope of 3/4 (negative reciprocal of -4/3). Using the point-slope form:

y - 1 = (3/4)(x - 6)

Simplifying, we get:

y = (3/4)x - 3.5

Advanced Concepts and Further Exploration

This guide provides a solid foundation for understanding the slope-intercept form. For more advanced exploration, you can delve into:

• Linear inequalities: Instead of an equation, you'll have an inequality (e.g., y > (-4/3)x + 4), representing a region on the graph.

• Systems of linear equations: Solving for the intersection point of two or more lines.

• Matrices and vectors: Representing linear equations and transformations using matrices.

• Multivariate linear regression: Extending the concept to equations with more than two variables.

Conclusion

Converting the equation 4x + 3y = 12 into slope-intercept form is a straightforward yet fundamental process in algebra. Mastering this skill is crucial for understanding linear relationships, graphing lines, and solving various problems in mathematics, science, and other disciplines. This guide provides a comprehensive overview of the process, its interpretations, applications, and related concepts, equipping you with the knowledge to tackle various linear equation problems confidently. Remember to practice regularly to solidify your understanding and develop proficiency in solving diverse problems involving linear equations.