3x 2y 6 Solve For Y

Solving for 'y': A Deep Dive into 3x + 2y = 6

This comprehensive guide explores the algebraic equation 3x + 2y = 6, demonstrating how to solve for 'y' and providing a detailed understanding of the underlying mathematical concepts. We'll cover various approaches, discuss the significance of this type of equation in different contexts, and offer practical examples to solidify your understanding. This article is designed to be helpful for students of algebra, those brushing up on their math skills, and anyone curious about the process of solving linear equations.

Understanding Linear Equations

Before diving into the solution, let's establish a solid foundation. The equation 3x + 2y = 6 is a linear equation in two variables, 'x' and 'y'. This means that when graphed, it represents a straight line. Linear equations are fundamental in mathematics and have countless applications in various fields, including physics, engineering, economics, and computer science.

Key characteristics of linear equations:

• Degree: The highest power of any variable is 1.
• Variables: They involve two or more variables, usually represented by letters like x, y, z, etc.
• Graph: Their graph is always a straight line.

The equation 3x + 2y = 6 represents a relationship between 'x' and 'y'. For any given value of 'x', there's a corresponding value of 'y' that satisfies the equation. Conversely, for any given 'y', there's a corresponding 'x'. Solving for 'y' means isolating 'y' on one side of the equation, expressing it in terms of 'x'.

Solving for 'y' in 3x + 2y = 6: Step-by-Step Approach

The goal is to manipulate the equation algebraically to isolate 'y'. Here's the step-by-step process:

Step 1: Isolate the term containing 'y'.

To isolate the term with 'y' (which is 2y), we need to move the term with 'x' (3x) to the other side of the equation. We achieve this by subtracting 3x from both sides:

3x + 2y - 3x = 6 - 3x

This simplifies to:

2y = 6 - 3x

Step 2: Solve for 'y'.

The 'y' term is currently multiplied by 2. To isolate 'y', we need to divide both sides of the equation by 2:

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

This simplifies to:

y = (6 - 3x)/2

Step 3: Simplify (Optional).

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

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

This simplifies to:

y = 3 - (3/2)x

Or, more commonly written as:

y = 3 - 1.5x

This is the final solution, expressing 'y' explicitly in terms of 'x'. This equation tells us that for any given value of x, we can calculate the corresponding value of y.

Understanding the Solution: Implications and Interpretations

The equation y = 3 - 1.5x represents a straight line when graphed. The slope of the line is -1.5 (the coefficient of x), and the y-intercept is 3 (the constant term).

• Slope: The slope of -1.5 indicates that for every one-unit increase in x, y decreases by 1.5 units. This signifies a negative linear relationship between x and y.

• Y-intercept: The y-intercept of 3 indicates that when x = 0, y = 3. This is the point where the line crosses the y-axis.

• X-intercept: To find the x-intercept (where the line crosses the x-axis, where y=0), we set y = 0 and solve for x:

0 = 3 - 1.5x 1.5x = 3 x = 2

The x-intercept is 2. This means the line crosses the x-axis at the point (2, 0).

Practical Applications and Examples

Linear equations like 3x + 2y = 6 have numerous real-world applications. Let's explore a few examples:

Example 1: Cost Calculation

Imagine you're running a small business. Your costs consist of a fixed cost (e.g., rent) and a variable cost (e.g., materials per unit produced). Let's say your fixed cost is $6, the variable cost per unit is $1.50, and 'x' represents the number of units produced. The total cost ('y') can be represented by the equation:

y = 1.5x + 6

This equation is similar in structure to our solved equation (though the variables have different meanings).

Example 2: Mixing Solutions

Suppose you're mixing two solutions with different concentrations. Let's say you have a 3% solution ('x') and a 6% solution ('y'), and you want to create 6 liters of a 2% solution. This scenario can be modeled using a linear equation:

0.03x + 0.06y = 0.02(6)

Solving this equation for 'y' will tell you how much of the 6% solution you need to mix with a given amount of the 3% solution to achieve the desired concentration.

Example 3: Distance-Time Problems

Consider a scenario involving constant speed. If you travel at a speed of 1.5 units per time unit (m/s for example) and have already traveled 3 units, your total distance (y) after a certain amount of time (x) can be represented by:

y = 1.5x + 3

Again, a similar structure to our solved equation, showcasing the versatility of linear equations.

Alternative Methods for Solving

While the method outlined above is straightforward, there are alternative approaches to solve for 'y' in 3x + 2y = 6:

1. Using a Graphing Calculator or Software: Graphing tools can easily plot the line represented by the equation 3x + 2y = 6. The graph visually shows the relationship between x and y, and you can determine the value of y for any given x.

2. Substitution Method: If you have another linear equation involving x and y (a system of equations), you can use the substitution method. Solve one equation for x (or y) and substitute that expression into the other equation.

3. Elimination Method: Similar to substitution, this method is used when you have a system of two linear equations. You manipulate the equations to eliminate one variable (either x or y) and then solve for the remaining variable.

Conclusion: Mastering Linear Equations

Solving for 'y' in the equation 3x + 2y = 6 is a fundamental algebraic skill with broad applications. Understanding the process, interpreting the results (slope, intercepts), and applying it to real-world scenarios is crucial for anyone studying mathematics or using mathematical models in their field. This detailed guide provided a comprehensive approach, alternative methods, and practical examples to solidify your understanding of this important concept. Remember to practice regularly to build your confidence and proficiency in solving linear equations and other algebraic problems. Further exploration into systems of linear equations and more complex algebraic concepts will build upon this foundational knowledge.