X 3y 12 In Slope Intercept Form

Transforming 𝑥 + 3𝑦 = 12 into Slope-Intercept Form: A Comprehensive Guide

The equation 𝑥 + 3𝑦 = 12 represents a linear relationship between two variables, 𝑥 and 𝑦. While this equation is in standard form (Ax + By = C), it's often more useful to express it in slope-intercept form (𝑦 = 𝑚𝑥 + 𝑏), where 'm' represents the slope and 'b' represents the y-intercept. This form allows for easy visualization of the line on a graph and simplifies calculations involving the line's characteristics. This guide will walk you through the step-by-step process of converting 𝑥 + 3𝑦 = 12 into slope-intercept form, exploring the meaning of the slope and y-intercept, and offering further applications and related concepts.

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 dependent variable, the value that changes based on the value of x.
• x: Represents the independent variable, the value that is chosen or controlled.
• m: Represents the slope of the line. The slope indicates the steepness and direction of the line. A positive slope signifies an upward trend (from left to right), while a negative slope signifies 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, the point where the line crosses the y-axis (where x = 0).

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

The conversion involves isolating 'y' on one side of the equation. Let's break down the steps:

1. Subtract x from both sides: This step aims to move the 'x' term to the right side of the equation. 𝑥 + 3𝑦 - 𝑥 = 12 - 𝑥 This simplifies to: 3𝑦 = -𝑥 + 12

2. Divide both sides by 3: This step isolates 'y' by dividing all terms by the coefficient of 'y', which is 3. (3𝑦)/3 = (-𝑥 + 12)/3 This simplifies to: 𝑦 = (-1/3)𝑥 + 4

Now, we have successfully converted the equation from standard form to slope-intercept form: 𝑦 = (-1/3)𝑥 + 4

Interpreting the Slope and Y-Intercept

From the slope-intercept form, 𝑦 = (-1/3)𝑥 + 4, we can directly extract valuable information about the line:

• Slope (m) = -1/3: This negative slope indicates that the line slopes downward from left to right. The value -1/3 tells us that for every 3 units increase in x, y decreases by 1 unit. This represents the rate of change between x and y.

• Y-intercept (b) = 4: This signifies that the line intersects the y-axis at the point (0, 4). This is the value of y when x is 0.

Graphing the Equation

With the slope-intercept form, graphing the line becomes straightforward:

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

2. Use the slope to find another point: Since the slope is -1/3, from the y-intercept (0, 4), move 3 units to the right (positive x-direction) and 1 unit down (negative y-direction). This gives you a second point (3, 3).

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

Further Applications and Related Concepts

Understanding the slope-intercept form extends beyond simple graphing. It's crucial in various mathematical and real-world applications:

1. Finding x-intercept:

The x-intercept is the point where the line crosses the x-axis (where y = 0). To find it, set y = 0 in the slope-intercept equation:

0 = (-1/3)x + 4

Solving for x:

x = 12

Therefore, the x-intercept is (12, 0).

2. Determining Parallel and Perpendicular Lines:

• Parallel Lines: Parallel lines have the same slope. Any line with a slope of -1/3 will be parallel to the line represented by 𝑦 = (-1/3)𝑥 + 4.

• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -1/3 is 3. Therefore, any line with a slope of 3 will be perpendicular to the line represented by 𝑦 = (-1/3)𝑥 + 4.

3. Real-world Applications:

The slope-intercept form finds extensive use in modeling real-world scenarios. For instance:

• Cost Functions: In economics, the equation could represent a cost function where x represents the number of units produced and y represents the total cost. The y-intercept would be the fixed cost, and the slope would be the variable cost per unit.

• Speed and Distance: The equation might model the relationship between time (x) and distance traveled (y) at a constant speed, where the slope represents the speed.

• Temperature Conversion: Linear equations can be used to convert between different temperature scales (e.g., Celsius and Fahrenheit), with the slope and y-intercept reflecting the conversion factors.

4. Solving Systems of Linear Equations:

The slope-intercept form is particularly useful when solving systems of linear equations graphically. By graphing multiple equations in slope-intercept form, the point of intersection (if it exists) represents the solution to the system.

Conclusion

Converting the equation 𝑥 + 3𝑦 = 12 into slope-intercept form, 𝑦 = (-1/3)𝑥 + 4, provides a clear and concise representation of the linear relationship between x and y. Understanding the slope and y-intercept allows for easy graphing and interpretation of the line's characteristics. This form is fundamental in various mathematical applications and real-world problem-solving, extending its significance beyond basic algebra. Its application in parallel and perpendicular line identification, x-intercept calculation, and real-world modeling showcases its versatility and importance in mathematics and beyond. Mastering this conversion is a cornerstone of understanding linear equations and their applications.