3x 4y 12 Solve For Y

Solving for 'y': A Comprehensive Guide to 3x + 4y = 12

This article provides a detailed, step-by-step guide on how to solve the equation 3x + 4y = 12 for 'y'. We'll explore various approaches, including algebraic manipulation and graphical representation, catering to different levels of mathematical understanding. We'll also discuss the concept of dependent and independent variables, and how this equation can be applied in real-world scenarios. This in-depth analysis aims to offer a comprehensive understanding of this linear equation and its implications.

Understanding the Equation: 3x + 4y = 12

The equation 3x + 4y = 12 is a linear equation in two variables, 'x' and 'y'. A linear equation represents a straight line when graphed. In this equation:

• 3x and 4y are terms involving the variables. The numbers 3 and 4 are called coefficients.
• 12 is the constant term.
• x and y are the variables representing unknown values. We aim to express 'y' in terms of 'x'.

This equation signifies a relationship between 'x' and 'y', where any changes in the value of 'x' will consequently affect the value of 'y' and vice-versa.

Solving for 'y' Algebraically

The primary goal is to isolate 'y' on one side of the equation. This involves a series of algebraic manipulations, adhering to the fundamental principles of equality: whatever operation you perform on one side of the equation, you must perform the same operation on the other side.

Here's the step-by-step algebraic solution:

1. Subtract 3x from both sides: This removes the 'x' term from the left side, leaving only the 'y' term.

   3x + 4y - 3x = 12 - 3x

   This simplifies to:

   4y = 12 - 3x

2. Divide both sides by 4: This isolates 'y', giving us the solution for 'y' in terms of 'x'.

   (4y)/4 = (12 - 3x)/4

   This simplifies to:

   y = 3 - (3/4)x

This is the solution. It expresses 'y' as a function of 'x'. For any given value of 'x', you can substitute it into this equation to find the corresponding value of 'y'.

Understanding the Solution: 'y' as a Function of 'x'

The equation y = 3 - (3/4)x represents a linear function. This means that the relationship between x and y is linear and can be visualized as a straight line on a graph.

• Independent Variable (x): 'x' is the independent variable because its value can be chosen freely.
• Dependent Variable (y): 'y' is the dependent variable because its value depends on the value chosen for 'x'.

The equation shows that as 'x' increases, 'y' decreases. The slope of the line represented by this equation is -3/4, indicating a negative correlation between 'x' and 'y'. The y-intercept is 3, meaning the line intersects the y-axis at the point (0, 3).

Graphical Representation

Plotting the equation on a graph provides a visual representation of the relationship between 'x' and 'y'. To do this, we can choose a few values for 'x', calculate the corresponding values for 'y' using the equation y = 3 - (3/4)x, and then plot these points on a coordinate plane.

Let's choose some values for x:

• If x = 0: y = 3 - (3/4)*0 = 3
• If x = 4: y = 3 - (3/4)*4 = 0
• If x = -4: y = 3 - (3/4)*(-4) = 6
• If x = 8: y = 3 - (3/4)*8 = -3

These points (0, 3), (4, 0), (-4, 6), and (8, -3) can be plotted on a graph. Connecting these points will reveal a straight line, confirming the linear nature of the equation.

Real-World Applications

Linear equations like 3x + 4y = 12 find applications in various real-world scenarios. For instance:

• Cost Calculations: Imagine 'x' represents the number of units of product A costing $3 each, and 'y' represents the number of units of product B costing $4 each. The equation shows that if you spend $12 in total, various combinations of 'x' and 'y' are possible.

• Resource Allocation: 'x' could represent the number of hours spent on task A, and 'y' could represent the number of hours spent on task B. If a total of 12 hours are available, this equation helps determine feasible allocations.

• Mixture Problems: In chemistry or cooking, the equation could represent a mixture of two ingredients with different weights or volumes.

• Economics: Linear equations are frequently used to model supply and demand relationships, where x might represent price and y represents quantity.

The versatility of linear equations makes them a fundamental tool in various fields.

Solving for 'x'

While the primary focus was solving for 'y', let's briefly show how to solve for 'x'. We start again with the original equation: 3x + 4y = 12.

1. Subtract 4y from both sides:

   3x + 4y - 4y = 12 - 4y

   This simplifies to:

   3x = 12 - 4y

2. Divide both sides by 3:

   (3x)/3 = (12 - 4y)/3

   This simplifies to:

   x = 4 - (4/3)y

This equation now expresses 'x' as a function of 'y'.

Further Exploration: Systems of Equations

The equation 3x + 4y = 12 is often part of a larger system of equations. Solving a system of equations involves finding values of 'x' and 'y' that satisfy all equations in the system simultaneously. Common methods for solving systems of equations include substitution and elimination.

Conclusion

Solving the equation 3x + 4y = 12 for 'y' involves straightforward algebraic manipulation. Understanding the solution as a linear function, representing a straight line on a graph, allows for a clearer grasp of the relationship between the variables. The algebraic solution, y = 3 - (3/4)x, is pivotal for understanding the equation's behavior and its applicability in various real-world contexts. This comprehensive guide emphasizes the importance of understanding both the algebraic and graphical interpretations of linear equations, empowering readers with a robust foundation in this fundamental mathematical concept. Remember to practice solving different linear equations to reinforce your understanding and improve your problem-solving skills. The ability to manipulate and interpret linear equations is crucial across various academic and professional fields.