4x 5 7 4y Solve For Y

Solving for y: A Comprehensive Guide to 4x + 5 = 7 + 4y

This article provides a detailed walkthrough of solving the algebraic equation 4x + 5 = 7 + 4y for the variable 'y'. We'll cover the fundamental principles of algebraic manipulation, step-by-step solutions, and practical applications. Understanding how to solve this type of equation is crucial for various mathematical concepts and real-world problem-solving.

Understanding the Equation

The equation 4x + 5 = 7 + 4y is a linear equation with two variables, x and y. 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'. The equation represents a straight line on a graph, and solving for 'y' gives us the equation in slope-intercept form (y = mx + b), where 'm' is the slope and 'b' is the y-intercept.

Step-by-Step Solution: Isolating 'y'

To solve for 'y', we need to manipulate the equation using algebraic operations. The goal is to get '4y' by itself on one side of the equals sign. Let's follow these steps:

Step 1: Subtract 7 from both sides

Subtracting 7 from both sides maintains the equality of the equation. This simplifies the equation, moving the constant term away from the term containing 'y'.

4x + 5 - 7 = 7 + 4y - 7

This simplifies to:

4x - 2 = 4y

Step 2: Divide both sides by 4

To isolate 'y', we need to divide both sides of the equation by the coefficient of 'y', which is 4.

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

This simplifies to:

x - 1/2 = y

Step 3: Rearrange the equation

Conventionally, we write the equation with 'y' on the left-hand side:

y = x - 1/2

Therefore, the solution for 'y' is y = x - 1/2

This means that for any given value of 'x', you can calculate the corresponding value of 'y' using this equation.

Verification: Checking the Solution

It's crucial to verify the solution to ensure accuracy. We can do this by substituting the solved equation for 'y' back into the original equation and checking if both sides are equal.

Let's substitute y = x - 1/2 into the original equation 4x + 5 = 7 + 4y:

4x + 5 = 7 + 4(x - 1/2)

Now, let's simplify the right-hand side:

4x + 5 = 7 + 4x - 2

4x + 5 = 4x + 5

Both sides are equal, confirming that our solution, y = x - 1/2, is correct.

Understanding the Solution: Graphical Representation

The equation y = x - 1/2 represents a straight line on a Cartesian coordinate system. The slope of the line is 1, and the y-intercept is -1/2. This means the line crosses the y-axis at the point (0, -1/2) and has a positive slope, indicating that it rises from left to right.

Plotting points and connecting them will visually represent the line defined by this equation. For instance, when x = 0, y = -1/2; when x = 1, y = 1/2; when x = 2, y = 3/2, and so on. Each of these points (x,y) satisfies the original equation 4x + 5 = 7 + 4y.

Applications and Extensions

Solving linear equations like 4x + 5 = 7 + 4y has numerous applications in various fields:

1. Physics and Engineering

Linear equations are frequently used to model physical phenomena and relationships between variables. For example, in physics, this type of equation could represent the relationship between distance, speed, and time, or the relationship between force, mass, and acceleration.

2. Economics and Finance

In economics, linear equations can be used to model supply and demand curves, budget constraints, or other economic relationships. In finance, they might be used to model relationships between interest rates, investment amounts, and returns.

3. Computer Science and Programming

Linear equations are fundamental in computer graphics, algorithms, and data analysis. They are used for tasks such as calculating coordinates, transforming objects, and representing data relationships.

4. Real-World Problem Solving

Many real-world problems can be modeled and solved using linear equations. For example, calculating the cost of a product based on variable costs and fixed costs, determining the amount of ingredients needed for a recipe, or calculating travel time based on distance and speed.

Solving Similar Equations: Variations and Challenges

The techniques used to solve 4x + 5 = 7 + 4y can be extended to solve other similar linear equations. The core principles remain the same:

• Isolate the variable: Your goal is always to isolate the variable you're solving for (in this case, 'y').
• Use inverse operations: Use addition, subtraction, multiplication, and division to manipulate the equation. Remember to perform the same operation on both sides to maintain equality.
• Simplify the equation: Combine like terms and simplify the equation as much as possible.
• Verify the solution: Always check your solution by substituting it back into the original equation.

Challenges: You might encounter equations with fractions, decimals, or negative numbers. However, the fundamental principles remain unchanged. You might need to employ additional steps like finding a common denominator when dealing with fractions or using the distributive property when dealing with parentheses.

Example: Consider the equation 2(x + 3) = 6y - 4. To solve for y, you would first distribute the 2, then add 4 to both sides, and finally divide by 6.

Conclusion: Mastering Algebraic Manipulation

Solving for 'y' in the equation 4x + 5 = 7 + 4y is a fundamental skill in algebra. By understanding the steps involved and the underlying principles, you can confidently solve a wide range of similar equations and apply this knowledge to various mathematical and real-world problems. Remember to practice regularly to solidify your understanding and improve your problem-solving skills. The ability to manipulate algebraic equations effectively is a crucial skill for success in many academic and professional fields. Consistent practice and careful attention to detail will lead to mastery.