Solve For Y 3x 2y 6

Solve for y: 3x + 2y = 6 – A Comprehensive Guide

Solving algebraic equations is a fundamental skill in mathematics, crucial for various applications in science, engineering, and everyday problem-solving. This article provides a comprehensive guide on how to solve the equation 3x + 2y = 6 for y, explaining the steps involved, providing variations, and exploring the broader context of linear equations.

Understanding the Equation: 3x + 2y = 6

The equation 3x + 2y = 6 is a linear equation in two variables, x and y. A linear equation represents a straight line when graphed on a coordinate plane. Solving for y means isolating y on one side of the equation, expressing it in terms of x. This allows us to determine the value of y for any given value of x.

Step-by-Step Solution: Isolating y

To solve for y, we need to manipulate the equation using algebraic operations, ensuring that we perform the same operation on both sides of the equation to maintain balance.

1. Subtract 3x from both sides:

Our goal is to isolate the term containing y, which is 2y. We begin by subtracting 3x from both sides of the equation:

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

This simplifies to:

2y = 6 - 3x

2. Divide both sides by 2:

Now, we need to isolate y by dividing both sides of the equation by 2:

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

This simplifies to:

y = (6 - 3x) / 2

3. Simplify (Optional):

While the equation is now solved for y, we can further simplify the expression by distributing the division:

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

y = 3 - (3/2)x

This simplified form, y = 3 - (3/2)x or y = 3 - 1.5x, is often preferred for its clarity and ease of use when substituting values for x.

Understanding the Solution: What it Means

The solution, y = 3 - (3/2)x, represents a linear relationship between x and y. It tells us that for any given value of x, we can calculate the corresponding value of y. This equation can be used to create a table of values or to graph the line on a coordinate plane.

For example:

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

These points (0, 3), (2, 0), and (-2, 6) lie on the line represented by the equation 3x + 2y = 6.

Graphical Representation of the Solution

Graphing the equation provides a visual representation of the solution. Plotting the points calculated above and connecting them will result in a straight line. This line visually represents all possible pairs of (x, y) values that satisfy the original equation 3x + 2y = 6. The slope of the line is -3/2, indicating that for every increase of 2 units in x, y decreases by 3 units. The y-intercept is 3, meaning the line crosses the y-axis at the point (0, 3).

Variations and Extensions: Similar Equation Structures

The principles used to solve 3x + 2y = 6 can be applied to solve other similar linear equations. Consider these variations:

• ax + by = c: This is the general form of a linear equation in two variables. The solution for y will always involve isolating the by term and then dividing by b.

• Equations with Fractions: Equations containing fractions can be solved using the same principles, but it might be helpful to clear the fractions first by multiplying both sides of the equation by the least common multiple of the denominators.

• Equations with Parentheses: Equations with parentheses require simplifying the equation first by expanding the parentheses using the distributive property before proceeding with the steps to solve for y.

• Solving for x: The process of solving for x instead of y is very similar. You would isolate the x term and then solve for x in terms of y.

Applications of Linear Equations

Linear equations, and the ability to solve them, have wide-ranging applications in various fields:

• Physics: Linear equations are frequently used to model physical phenomena, such as motion, electricity, and magnetism.

• Engineering: Engineers use linear equations to design structures, circuits, and systems.

• Economics: Linear equations are used to model economic relationships, such as supply and demand.

• Computer Science: Linear equations are used in computer graphics, machine learning, and optimization algorithms.

Practical Examples: Real-world Applications

Let’s look at a couple of real-world examples illustrating the application of solving for y in a linear equation:

Example 1: Calculating Costs

Suppose you are renting a car. The rental company charges a fixed fee of $6 plus $3 per hour. The total cost (y) can be represented by the equation: 3x + y = 6, where x is the number of hours rented. Solving for y gives you y = 6 - 3x, allowing you to easily calculate the total cost for any number of rental hours.

Example 2: Mixing Solutions

Imagine you’re mixing two solutions. You need a final volume of 6 liters, with one solution having 3 times the concentration of the other. Let x represent the volume of the less concentrated solution, and y the volume of the more concentrated solution. The equation would be 3x + y = 6. Solving for y (y = 6 - 3x) allows you to determine the volume of the more concentrated solution needed for any given volume of the less concentrated solution.

Troubleshooting Common Mistakes

When solving linear equations, several common mistakes can occur:

• Incorrectly applying the order of operations: Remember to follow the order of operations (PEMDAS/BODMAS) correctly.

• Making errors in arithmetic: Double-check your calculations to ensure accuracy.

• Forgetting to perform the same operation on both sides: Always perform the same operation on both sides of the equation to maintain balance.

Conclusion: Mastering Linear Equations

Solving the equation 3x + 2y = 6 for y, which yields y = 3 - (3/2)x, provides a solid foundation for understanding linear equations and their applications. Mastering this process is essential for more advanced mathematical concepts and problem-solving in various fields. By carefully following the steps outlined and practicing with different variations of linear equations, you can develop proficiency in solving for variables and confidently apply this skill in various real-world situations. Remember to always check your work and practice regularly to build confidence and accuracy.