3x 2y 6 Solve For Y

Solving for 'y': A Comprehensive Guide to 3x + 2y = 6

This article provides a detailed walkthrough of how to solve the equation 3x + 2y = 6 for 'y', exploring various methods and their applications. We'll delve into the fundamental algebraic principles involved, discuss different approaches to reach the solution, and highlight the importance of understanding the underlying concepts. This comprehensive guide is designed for students, educators, and anyone looking to strengthen their algebraic skills.

Understanding the Equation: 3x + 2y = 6

The equation 3x + 2y = 6 represents a linear equation in two variables, 'x' and 'y'. A linear equation, in its simplest form, depicts a straight line when graphed on a Cartesian coordinate system. This equation indicates a relationship where the values of 'x' and 'y' are interconnected; a change in one variable directly affects the other. Our goal is to isolate 'y' to express it solely in terms of 'x', revealing this relationship explicitly.

Method 1: Isolating 'y' Through Algebraic Manipulation

This is the most straightforward approach. We'll use basic algebraic operations to rearrange the equation and solve for 'y':

1. Subtract 3x from both sides:

The initial equation is: 3x + 2y = 6

Subtracting 3x from both sides maintains the equation's balance:

2y = 6 - 3x

2. Divide both sides by 2:

To isolate 'y', we divide both sides of the equation by 2:

y = (6 - 3x) / 2

3. Simplify (Optional):

While the above equation is perfectly correct, we can simplify it further by distributing the division:

y = 3 - (3/2)x

Therefore, the solution for 'y' is y = 3 - (3/2)x or equivalently y = (6 - 3x) / 2. This equation tells us that for any given value of 'x', we can calculate the corresponding value of 'y' that satisfies the original equation.

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

The slope-intercept form of a linear equation, y = mx + c, provides a clear representation of the line's slope ('m') and y-intercept ('c'). We can rewrite our equation in this form:

1. Rearrange the Equation:

Starting with 3x + 2y = 6, we follow the same steps as in Method 1 to isolate 'y':

2y = 6 - 3x

y = (6 - 3x) / 2

y = 3 - (3/2)x

2. Identify the Slope and y-intercept:

Comparing this equation to y = mx + c, we identify:

m (slope) = -3/2: This indicates the line's steepness and direction. A negative slope signifies a downward-sloping line.
c (y-intercept) = 3: This is the point where the line intersects the y-axis (when x = 0).

This form provides a concise and easily interpretable representation of the linear relationship between 'x' and 'y'.

Method 3: Graphical Representation

Visualizing the equation graphically reinforces understanding. Plotting the equation on a Cartesian plane helps to see the relationship between 'x' and 'y':

1. Find the Intercepts:

To plot the line, we find the x-intercept (where y = 0) and the y-intercept (where x = 0).

y-intercept: Setting x = 0 in the original equation (3x + 2y = 6) gives us 2y = 6, so y = 3. The y-intercept is (0, 3).
x-intercept: Setting y = 0 in the original equation gives us 3x = 6, so x = 2. The x-intercept is (2, 0).

2. Plot the Points and Draw the Line:

Plot the points (0, 3) and (2, 0) on the graph. Draw a straight line passing through these points. This line visually represents all possible (x, y) pairs that satisfy the equation 3x + 2y = 6. Any point on this line will provide a valid solution for 'x' and 'y'.

Applications and Significance

Solving for 'y' in equations like 3x + 2y = 6 has broad applications across various fields:

Economics: Analyzing supply and demand curves, modeling economic relationships.
Physics: Representing linear relationships between physical quantities like velocity, acceleration, and time.
Engineering: Modeling linear systems, analyzing circuits, and designing structures.
Computer Science: Solving systems of linear equations in algorithms and simulations.
Statistics: Analyzing regression lines and interpreting data relationships.

Understanding how to manipulate and solve linear equations is a fundamental skill in mathematics and has far-reaching implications in many practical applications.

Advanced Concepts and Extensions

While we focused on solving for 'y' in a simple linear equation, the principles extend to more complex scenarios:

Systems of Linear Equations: Solving for multiple variables in multiple equations requires techniques like substitution or elimination.
Non-Linear Equations: Equations involving exponents, roots, or other non-linear functions require different solution methods.
Calculus: Derivatives and integrals are used to analyze the rates of change and areas under curves described by more complex equations.

Mastering the fundamentals of solving linear equations is crucial for tackling these more advanced mathematical concepts.

Practical Exercises

To solidify your understanding, try solving for 'y' in these similar equations:

2x + 4y = 8
5x - y = 10
-x + 3y = 9

Compare your solutions with the methods explained above. Check your answers by substituting your solution for 'y' back into the original equation to ensure it holds true.

Conclusion: Mastering the Fundamentals

Solving for 'y' in the equation 3x + 2y = 6 is a fundamental algebraic skill with wide-ranging applications. Through algebraic manipulation, using the slope-intercept form, or visualising the solution graphically, we've explored different approaches to find the solution: y = 3 - (3/2)x. Understanding this seemingly simple equation lays a solid foundation for tackling more complex mathematical problems in various fields. Consistent practice and a strong grasp of the underlying concepts are key to mastering algebraic skills and applying them effectively. Remember to always check your work by substituting your solution back into the original equation to verify its accuracy. This diligent approach will enhance your confidence and proficiency in algebra.