Solve For Y 2x 3y 12

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

Solving algebraic equations is a fundamental skill in mathematics, crucial for various applications from simple calculations to complex scientific modeling. This article focuses on solving the linear equation 2x + 3y = 12 for y, providing a detailed explanation of the process and exploring related concepts. We'll delve into the step-by-step solution, examine the significance of the equation's form, explore its graphical representation, and discuss how to solve similar equations with varying complexity. Finally, we'll touch upon applications of this fundamental skill in different fields.

Understanding the Equation: 2x + 3y = 12

The equation 2x + 3y = 12 represents a linear relationship between two variables, x and y. This means that when plotted on a graph, it forms a straight line. The equation is in its standard form, Ax + By = C, where A, B, and C are constants. Our goal is to isolate y, expressing it as a function of x. This means we want to rearrange the equation to get y = f(x), where f(x) is an expression involving x.

Step-by-Step Solution: Isolating y

To solve for y, we need to manipulate the equation using algebraic operations, ensuring we maintain equality at each step. Here's the step-by-step process:

1. Subtract 2x from both sides: This eliminates 2x from the left side, leaving only terms involving y. 2x + 3y - 2x = 12 - 2x This simplifies to: 3y = 12 - 2x

2. Divide both sides by 3: This isolates y by removing the coefficient of 3. 3y / 3 = (12 - 2x) / 3 This simplifies to: y = (12 - 2x) / 3

3. Simplify (Optional): We can further simplify the expression by distributing the division: y = 12/3 - (2x)/3 y = 4 - (2/3)x or equivalently, y = - (2/3)x + 4

This final form, y = -(2/3)x + 4, represents the equation in slope-intercept form (y = mx + b), where m is the slope (-2/3) and b is the y-intercept (4).

Graphical Representation and Interpretation

The equation y = -(2/3)x + 4 represents a straight line on a Cartesian coordinate system. The y-intercept, 4, indicates that the line crosses the y-axis at the point (0, 4). The slope, -2/3, indicates that for every 3 units increase in x, y decreases by 2 units. This negative slope signifies a downward trend of the line. Plotting several points satisfying the equation and connecting them will reveal this straight line. Understanding the graphical representation helps visualize the relationship between x and y.

Finding points on the line:

To plot the line, we can find several points that satisfy the equation. For instance:

• If x = 0, then y = 4 (0, 4)
• If x = 3, then y = 2 (3, 2)
• If x = 6, then y = 0 (6, 0)
• If x = -3, then y = 6 (-3, 6)

These points, when plotted and connected, will accurately represent the line described by the equation.

Solving Similar Equations: Variations and Techniques

The method used to solve 2x + 3y = 12 for y can be applied to other linear equations. Let's explore some variations:

Example 1: 5x - 2y = 10

1. Add 2y to both sides: 5x = 10 + 2y
2. Subtract 10 from both sides: 5x - 10 = 2y
3. Divide by 2: y = (5x - 10) / 2 or y = (5/2)x - 5

Example 2: x + y = 7

This is a simpler case:

1. Subtract x from both sides: y = 7 - x

Example 3: Equations with Fractions

Consider the equation: (1/2)x + (2/3)y = 1

1. Subtract (1/2)x: (2/3)y = 1 - (1/2)x
2. Multiply by the reciprocal of (2/3): y = (3/2)(1 - (1/2)x)
3. Simplify: y = (3/2) - (3/4)x

These examples highlight the adaptability of the basic algebraic principles used to solve for y in a linear equation. The key steps remain consistent: isolate the term containing y, and then solve for y using appropriate arithmetic operations.

Applications in Different Fields

Solving linear equations like 2x + 3y = 12 is not just a theoretical exercise; it has wide-ranging applications across numerous fields:

• Economics: Linear equations model supply and demand, cost functions, and other economic relationships. Solving for a particular variable allows economists to analyze market equilibrium and predict outcomes based on changes in other factors.

• Physics: Linear equations are crucial in mechanics, describing motion, forces, and energy. Solving for variables such as velocity or acceleration allows physicists to model and predict physical phenomena.

• Engineering: Engineers use linear equations extensively in structural analysis, circuit design, and many other areas. Solving for unknown quantities enables engineers to design and optimize systems.

• Computer Science: Linear equations form the basis of many algorithms used in computer graphics, machine learning, and data analysis.

• Finance: Linear equations are used in financial modeling to determine investment returns, calculate loan repayments, and analyze risk.

Conclusion: Mastering the Fundamentals

The ability to solve for y (or any variable) in a linear equation is a fundamental skill with far-reaching implications. The techniques demonstrated here – involving careful application of algebraic operations – are not only applicable to simple equations like 2x + 3y = 12 but form the groundwork for solving more complex mathematical problems. Mastering this fundamental skill paves the way for success in more advanced mathematical concepts and applications across various disciplines. Continuous practice and understanding the underlying principles are key to developing proficiency in solving algebraic equations.