3x 2y 12 Solve For Y

Solving for 'y': A Comprehensive Guide to 3x + 2y = 12

The equation 3x + 2y = 12 represents a linear equation in two variables, x and y. Solving for 'y' means isolating 'y' on one side of the equation, expressing it in terms of 'x'. This process is fundamental in algebra and has numerous applications in various fields, from calculating slopes of lines to solving systems of equations and understanding relationships between variables. This comprehensive guide will walk you through various methods of solving for 'y' in 3x + 2y = 12, explaining the underlying principles and demonstrating practical applications.

Understanding the Equation: 3x + 2y = 12

Before diving into the solution, let's understand the equation itself. This equation represents a straight line on a Cartesian coordinate system. The values of 'x' and 'y' that satisfy the equation are the coordinates of points that lie on this line. The equation is in the standard form of a linear equation, Ax + By = C, where A, B, and C are constants. In our case, A = 3, B = 2, and C = 12.

Method 1: Isolating 'y' through Algebraic Manipulation

This is the most common and straightforward method. The goal is to isolate 'y' by performing inverse operations on both sides of the equation.

Steps:

1. Subtract 3x from both sides: This eliminates the 'x' term from the left side, leaving only the 'y' term. The equation becomes:

   2y = 12 - 3x

2. Divide both sides by 2: This isolates 'y', giving us the solution:

   y = (12 - 3x) / 2

   This can also be written as:

   y = 6 - (3/2)x

   This is the solution for 'y' in terms of 'x'. For any given value of 'x', you can substitute it into this equation to find the corresponding value of 'y'.

Example:

Let's find the value of 'y' when x = 2:

y = 6 - (3/2) * 2 = 6 - 3 = 3

Therefore, when x = 2, y = 3. The point (2, 3) lies on the line represented by the equation 3x + 2y = 12.

Method 2: Using the Slope-Intercept Form (y = mx + b)

The slope-intercept form of a linear equation is y = mx + b, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis). We can rearrange our equation to fit this form. This method offers a clearer understanding of the line's characteristics.

Steps:

1. Start with the original equation: 3x + 2y = 12

2. Subtract 3x from both sides: 2y = -3x + 12

3. Divide both sides by 2: y = (-3/2)x + 6

Now the equation is in slope-intercept form. The slope (m) is -3/2, and the y-intercept (b) is 6. This tells us that the line slopes downwards (negative slope) and crosses the y-axis at the point (0, 6).

Method 3: Graphical Representation

While not directly "solving" for 'y' algebraically, a graphical representation provides a visual understanding of the solution.

Steps:

1. Find the x-intercept: Set y = 0 and solve for x: 3x + 2(0) = 12 => x = 4. The x-intercept is (4, 0).

2. Find the y-intercept: Set x = 0 and solve for y: 3(0) + 2y = 12 => y = 6. The y-intercept is (0, 6).

3. Plot the intercepts: Plot the points (4, 0) and (0, 6) on a Cartesian coordinate system.

4. Draw the line: Draw a straight line through the two plotted points. This line represents all the points (x, y) that satisfy the equation 3x + 2y = 12.

Any point on this line represents a solution pair (x, y) to the equation. While this doesn't give an explicit formula for 'y' in terms of 'x', it visually demonstrates the relationship between the two variables.

Applications of Solving for 'y'

Solving for 'y' is a crucial skill with broad applications across various fields:

• Economics: Analyzing supply and demand curves, where 'x' might represent price and 'y' represents quantity. Solving for 'y' allows economists to determine the quantity demanded or supplied at a given price.

• Physics: Modeling linear relationships between physical quantities, such as velocity and time, or force and acceleration.

• Computer Science: Developing algorithms and computer programs involving linear equations. Solving for 'y' helps in calculating specific values needed for program execution.

• Engineering: Solving for unknown variables in engineering design, such as calculating the dimensions of structures or analyzing electrical circuits.

• Statistics: Analyzing relationships between variables in statistical data, using linear regression to model trends and make predictions. Solving for 'y' allows for predicting the value of the dependent variable (y) given the independent variable (x).

• Calculus: Finding the slope of a tangent line to a curve at a given point, which involves finding the derivative of the function. The process often involves isolating 'y' first.

Solving Systems of Equations

The ability to solve for 'y' (or x) is crucial when dealing with systems of equations. For example, consider another equation: x + y = 5. To solve this system of two equations (3x + 2y = 12 and x + y = 5), we can use substitution or elimination methods. Solving for 'y' in one equation and substituting it into the other is a common approach.

Advanced Concepts and Extensions

The equation 3x + 2y = 12 represents a simple linear equation. However, the principles of solving for 'y' extend to more complex scenarios:

• Non-linear equations: Equations involving higher powers of x and y require different methods of solution, often involving techniques from calculus or numerical analysis.

• Systems of non-linear equations: These require advanced techniques, such as iterative methods or graphical analysis.

• Partial differential equations: These equations involve partial derivatives and are used to model complex systems in physics and engineering. Solving for a particular variable often involves integrating or applying transform methods.

Conclusion

Solving for 'y' in the equation 3x + 2y = 12, while seemingly a simple algebraic manipulation, is a fundamental skill with far-reaching applications. Understanding the different methods – algebraic manipulation, slope-intercept form, and graphical representation – provides a solid foundation for tackling more complex mathematical problems in various fields. Mastering this skill opens doors to deeper understanding and problem-solving capabilities in numerous academic and professional disciplines. The ability to isolate a variable and express it in terms of others is a cornerstone of mathematical modeling and analysis.