2 1 6y 3x 4 Solve For Y

Solving for 'y': A Comprehensive Guide to 2y + 3x = 4

This article provides a comprehensive guide on how to solve the equation 2y + 3x = 4 for 'y'. We'll explore various approaches, delve into the underlying mathematical concepts, and offer practical examples to solidify your understanding. This detailed explanation aims to be helpful for students of all levels, from those just beginning their algebra journey to those seeking a refresher. We'll also touch upon related concepts and applications to broaden your mathematical knowledge.

Understanding the Equation: 2y + 3x = 4

The equation 2y + 3x = 4 is a linear equation in two variables, x and y. A linear equation represents a straight line when graphed on a coordinate plane. The goal of "solving for y" is to isolate 'y' on one side of the equation, expressing it in terms of 'x'. This means we want to manipulate the equation algebraically until we have an equation in the form y = ...

Step-by-Step Solution: Isolating 'y'

To solve for 'y', we need to perform a series of algebraic manipulations. Here's a step-by-step breakdown:

1. Identify the term with 'y': In our equation, 2y is the term containing 'y'.

2. Isolate the term with 'y': We need to move the term with 'x' to the other side of the equation. To do this, we subtract 3x from both sides:

   2y + 3x - 3x = 4 - 3x

   This simplifies to:

   2y = 4 - 3x

3. Solve for 'y': Now, we need to isolate 'y' completely. Since 'y' is multiplied by 2, we divide both sides of the equation by 2:

   2y / 2 = (4 - 3x) / 2

   This gives us the solution:

   y = 2 - (3/2)x or equivalently, y = 2 - 1.5x

This final equation expresses 'y' explicitly in terms of 'x'. For any given value of 'x', you can substitute it into this equation to find the corresponding value of 'y'.

Graphical Representation

The equation y = 2 - (3/2)x represents a straight line. We can easily graph this line by finding two points that satisfy the equation and plotting them on a coordinate plane.

• If x = 0: y = 2 - (3/2)(0) = 2. So, one point is (0, 2).
• If x = 2: y = 2 - (3/2)(2) = 2 - 3 = -1. So, another point is (2, -1).

Plotting these two points and drawing a line through them gives us the graphical representation of the equation. This visual representation helps understand the relationship between x and y.

Checking the Solution

It's crucial to verify our solution. Let's choose a value for x, say x = 4, and substitute it into both the original equation and our solved equation:

• Original Equation: 2y + 3(4) = 4 => 2y + 12 = 4 => 2y = -8 => y = -4
• Solved Equation: y = 2 - (3/2)(4) = 2 - 6 = -4

Since both equations give the same value of y (-4) for x = 4, our solution is correct. This process of checking your solution is a vital part of problem-solving in mathematics.

Further Exploration: Related Concepts and Applications

Understanding the equation 2y + 3x = 4 extends beyond simply solving for 'y'. Let's explore some related concepts and applications:

1. Slope-Intercept Form: Our solved equation, y = 2 - (3/2)x, is in the slope-intercept form, y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept (the point where the line crosses the y-axis). In our equation:

• Slope (m) = -3/2: This indicates the line has a negative slope, meaning it slopes downwards from left to right.
• Y-intercept (b) = 2: This means the line crosses the y-axis at the point (0, 2).

Understanding the slope and intercept provides valuable information about the line's characteristics.

2. Systems of Linear Equations: The equation 2y + 3x = 4 can be part of a system of linear equations. A system of equations involves multiple equations with the same variables. Solving a system of equations means finding values of x and y that satisfy all equations simultaneously. For example, consider the system:

2y + 3x = 4 y - x = 1

We can use substitution or elimination methods to solve this system and find the unique solution (x, y) that satisfies both equations.

3. Real-World Applications: Linear equations like 2y + 3x = 4 have numerous real-world applications. For example, they can model:

• Cost functions: Where 'x' might represent the number of units produced, 'y' represents the total cost, and the equation describes the relationship between production and cost.
• Relationships between variables: In various scientific and engineering contexts, linear equations describe relationships between different quantities.
• Supply and Demand: In economics, linear equations can represent supply and demand curves.

4. Advanced Concepts: Solving for 'y' lays a foundation for more advanced mathematical concepts such as:

• Linear Transformations: These are mathematical operations that transform one vector into another.
• Matrix Algebra: Matrices can be used to represent and solve systems of linear equations efficiently.
• Calculus: The concepts of slope and derivatives build upon the understanding of linear equations.

Conclusion

Solving the equation 2y + 3x = 4 for 'y' is a fundamental algebraic skill. This detailed walkthrough demonstrated the step-by-step process, explained the underlying mathematical concepts, and explored related applications. By mastering this basic skill, you build a solid foundation for tackling more complex mathematical problems and understanding their real-world implications. Remember that practice is key to solidifying your understanding, so try solving similar equations and exploring different approaches to enhance your problem-solving abilities. The more you practice, the more confident and proficient you'll become in algebraic manipulation.