2x 3y 8 Solve For Y

Solving for 'y': A Comprehensive Guide to 2x + 3y = 8

This article provides a detailed explanation of how to solve the equation 2x + 3y = 8 for 'y', covering various approaches and their applications. We will delve into the fundamental algebraic principles involved, explore different methods of solution, and discuss the significance of this type of problem in various mathematical contexts. We'll also touch upon the broader implications of linear equations and their real-world applications.

Understanding the Equation: 2x + 3y = 8

The equation 2x + 3y = 8 is a linear equation in two variables, 'x' and 'y'. This means that when graphed, it represents a straight line. The equation shows a relationship where the sum of twice a value ('x') and three times another value ('y') equals 8. Our goal is to isolate 'y' on one side of the equation, expressing it in terms of 'x'. This process will provide us with the equation of the line in slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept.

Method 1: Isolating 'y' Through Algebraic Manipulation

This is the most straightforward method. We systematically manipulate the equation to isolate 'y' using basic algebraic operations.

Step-by-Step Solution:

1. Subtract 2x from both sides: This removes the '2x' term from the left side, leaving only the '3y' term. The equation becomes:

   3y = 8 - 2x

2. Divide both sides by 3: This isolates 'y' by eliminating the coefficient of 3. The equation now becomes:

   y = (8 - 2x) / 3

3. Simplify (Optional): While this form is perfectly acceptable, we can simplify it further by distributing the division:

   y = 8/3 - (2/3)x

   Therefore, the solution for y is y = 8/3 - (2/3)x or equivalently y = - (2/3)x + 8/3. This is the slope-intercept form, where the slope (m) is -2/3 and the y-intercept (b) is 8/3.

Method 2: Using the Properties of Equality

This method emphasizes the fundamental properties of equality – the ability to add, subtract, multiply, or divide both sides of an equation by the same value without changing the equality.

Step-by-Step Solution:

1. Apply the Subtraction Property of Equality: Subtract 2x from both sides of the equation:

   2x + 3y - 2x = 8 - 2x

   This simplifies to:

   3y = 8 - 2x

2. Apply the Division Property of Equality: Divide both sides of the equation by 3:

   3y / 3 = (8 - 2x) / 3

   This simplifies to:

   y = (8 - 2x) / 3 or y = 8/3 - (2/3)x

This method reinforces the underlying principles of algebraic manipulation and emphasizes the importance of maintaining equality throughout the process.

Method 3: Graphical Representation and Interpretation

While not directly solving for 'y' algebraically, a graphical representation provides valuable insights. Plotting the equation 2x + 3y = 8 on a Cartesian coordinate system visually demonstrates the relationship between 'x' and 'y'.

Understanding the Graph:

The graph of 2x + 3y = 8 is a straight line. Each point (x, y) on this line represents a solution to the equation. By observing the graph, we can see how 'y' changes as 'x' changes, reflecting the slope of the line. The y-intercept (where the line crosses the y-axis) represents the value of 'y' when x = 0.

Real-World Applications of Linear Equations

Linear equations like 2x + 3y = 8 are ubiquitous in various fields:

• Economics: Modeling supply and demand, calculating costs and profits, analyzing market trends.
• Physics: Describing motion, calculating forces, analyzing electrical circuits.
• Engineering: Designing structures, analyzing stresses and strains, controlling systems.
• Computer Science: Developing algorithms, creating simulations, modeling networks.
• Finance: Calculating interest, managing investments, forecasting financial performance.

Expanding on the Solution: Finding Specific Solutions

The solution y = 8/3 - (2/3)x allows us to find specific values of 'y' for given values of 'x'. For example:

• If x = 0: y = 8/3 - (2/3)(0) = 8/3
• If x = 3: y = 8/3 - (2/3)(3) = 8/3 - 2 = 2/3
• If x = -3: y = 8/3 - (2/3)(-3) = 8/3 + 2 = 14/3

Further Exploration: Systems of Linear Equations

The equation 2x + 3y = 8 is often part of a larger system of equations. Solving a system involves finding the values of 'x' and 'y' that satisfy all equations simultaneously. Common methods for solving systems of linear equations include:

• Substitution: Solving one equation for one variable and substituting it into the other equation.
• Elimination: Adding or subtracting the equations to eliminate one variable.
• Graphical Method: Finding the point of intersection of the lines representing the equations.

Conclusion: Mastering the Fundamentals

Solving for 'y' in the equation 2x + 3y = 8 is a fundamental algebraic skill with broad applications. Understanding the different methods—algebraic manipulation, using properties of equality, and graphical representation—provides a robust foundation for tackling more complex mathematical problems. Mastering this skill opens doors to more advanced concepts in mathematics, science, and engineering. The ability to solve linear equations is not only crucial for academic success but also valuable in many real-world situations. The steps involved in solving this seemingly simple equation highlight the power of basic algebraic principles and their widespread utility. Remember to always check your solutions by substituting the values of 'x' and 'y' back into the original equation to verify their accuracy. This practice reinforces understanding and helps prevent errors.