6x 3y 6 Solve For Y

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

This article provides a detailed, step-by-step guide on how to solve the equation 6x + 3y = 6 for 'y'. We'll explore various methods, delve into the underlying algebraic principles, and discuss practical applications. Understanding this seemingly simple equation is fundamental to mastering more complex algebraic concepts.

Understanding the Equation: 6x + 3y = 6

The equation 6x + 3y = 6 is a linear equation in two variables, 'x' and 'y'. This means that when graphed, it represents a straight line. Solving for 'y' means isolating 'y' on one side of the equation, expressing it in terms of 'x'. This gives us the equation of the line in slope-intercept form, y = mx + c, where 'm' is the slope and 'c' is the y-intercept.

Key Concepts to Remember

Before we begin solving, let's refresh some crucial algebraic concepts:

Inverse Operations: To isolate a variable, we perform the inverse (opposite) operation. Addition and subtraction are inverse operations, as are multiplication and division.
Order of Operations (PEMDAS/BODMAS): Remember to follow the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Distributive Property: This property states that a(b + c) = ab + ac. We might need to use this property in more complex variations of this equation.

Method 1: Solving for 'y' Step-by-Step

Let's solve the equation 6x + 3y = 6 for 'y' using a step-by-step approach:

Subtract 6x from both sides: This isolates the term containing 'y'.
```
6x + 3y - 6x = 6 - 6x
3y = 6 - 6x
```
Divide both sides by 3: This isolates 'y'.
```
3y / 3 = (6 - 6x) / 3
y = 2 - 2x
```

Therefore, the solution for 'y' is y = 2 - 2x. This is the slope-intercept form of the equation, where the slope (m) is -2 and the y-intercept (c) is 2.

Method 2: Rearranging the Equation

Another way to approach this is by rearranging the equation strategically:

Start with the original equation: 6x + 3y = 6
Notice the common factor: Observe that all terms are divisible by 3. Dividing the entire equation by 3 simplifies the equation:
```
(6x + 3y) / 3 = 6 / 3
2x + y = 2
```
Subtract 2x from both sides: This isolates 'y'.
```
2x + y - 2x = 2 - 2x
y = 2 - 2x
```

This method achieves the same result, demonstrating that simplifying the equation first can sometimes make the solving process easier.

Graphical Representation and Interpretation

The equation y = 2 - 2x represents a straight line. Let's analyze its graphical representation:

Slope (m) = -2: The negative slope indicates that the line descends from left to right. A slope of -2 means that for every 1 unit increase in x, y decreases by 2 units.
Y-intercept (c) = 2: The y-intercept is the point where the line crosses the y-axis. In this case, it crosses at the point (0, 2).
X-intercept: To find the x-intercept (where the line crosses the x-axis), we set y = 0 and solve for x:
```
0 = 2 - 2x
2x = 2
x = 1
```
The x-intercept is at the point (1, 0).

By plotting these points (0, 2) and (1, 0) and drawing a line through them, you can visualize the graphical representation of the equation 6x + 3y = 6.

Applications and Real-World Examples

Understanding how to solve for 'y' in linear equations like 6x + 3y = 6 has numerous practical applications across various fields:

Economics: Linear equations are used to model supply and demand. Solving for 'y' might represent finding the quantity demanded (y) given a certain price (x).
Physics: Linear equations describe relationships between physical quantities like distance, speed, and time. Solving for a specific variable allows for calculations and predictions.
Engineering: In engineering design, linear equations are used in numerous calculations, from structural analysis to circuit design. Solving for variables is crucial for determining optimal parameters.
Computer Science: Linear equations form the basis of many algorithms and calculations used in computer graphics, machine learning, and other areas.

Solving Variations of the Equation

Let's consider variations of the equation 6x + 3y = 6 and how to adapt our solving methods:

Scenario 1: 6x + 3y = 12

The only difference here is the constant term on the right-hand side. Following the same steps as before:

Subtract 6x: 3y = 12 - 6x
Divide by 3: y = 4 - 2x

The solution is y = 4 - 2x. Notice that the slope remains the same (-2), but the y-intercept changes to 4.

Scenario 2: 12x + 6y = 18

Here, the coefficients are different but maintain a consistent ratio. Simplifying first is advantageous:

Divide by 6: 2x + y = 3
Subtract 2x: y = 3 - 2x

The solution remains y = 3 - 2x. Again, the slope remains the same (-2), and only the y-intercept changes.

Scenario 3: Equations with Fractions or Decimals

Equations may involve fractions or decimals. In such cases, it is often easier to clear the fractions or decimals before solving. For example, if the equation were:

(1/2)x + (1/3)y = 1

You could multiply the entire equation by the least common multiple of the denominators (6) to eliminate the fractions before proceeding with the standard solving steps.

Conclusion: Mastering the Fundamentals

Solving the equation 6x + 3y = 6 for 'y', while seemingly simple, establishes a fundamental understanding of algebraic manipulation and linear equations. This knowledge serves as a building block for tackling more complex algebraic problems and finding practical applications across various disciplines. The methods discussed – step-by-step solving, rearranging the equation, and graphical interpretation – equip you with versatile tools to handle similar equations effectively. Remember to always practice and apply these concepts to strengthen your understanding. By mastering these fundamental techniques, you'll be well-prepared to tackle more advanced mathematical challenges.