What Is The Solution Of 3x 5 2x 7

Solving the Equation: 3x + 5 = 2x + 7 – A Step-by-Step Guide

This article provides a comprehensive walkthrough of solving the algebraic equation 3x + 5 = 2x + 7. While seemingly simple, this equation serves as a fundamental example illustrating core principles in algebra, applicable to more complex problems. We'll break down the solution step-by-step, explaining the underlying logic and techniques involved. We'll also explore related concepts and provide practice exercises to solidify your understanding.

Understanding the Equation

Before diving into the solution, let's analyze the equation: 3x + 5 = 2x + 7. This is a linear equation because the highest power of the variable 'x' is 1. The goal is to find the value of 'x' that makes the equation true – the value that makes the left-hand side (LHS) equal to the right-hand side (RHS).

The equation involves:

• Variables: 'x' represents an unknown value we need to determine.
• Constants: 5 and 7 are constant numerical values.
• Coefficients: 3 and 2 are the coefficients of the variable 'x'.

Solving the Equation: A Step-by-Step Approach

The fundamental principle in solving linear equations is to isolate the variable ('x' in this case) on one side of the equation. This involves performing the same operation on both sides to maintain the equation's balance. Here's the step-by-step solution:

Step 1: Gather the 'x' terms on one side.

To begin, we'll subtract 2x from both sides of the equation. This eliminates the 'x' term from the right side:

3x + 5 - 2x = 2x + 7 - 2x

This simplifies to:

x + 5 = 7

Step 2: Isolate the 'x' term.

Next, we need to isolate the 'x' term by removing the constant (+5) from the left side. We achieve this by subtracting 5 from both sides:

x + 5 - 5 = 7 - 5

This simplifies to:

x = 2

Step 3: Verify the Solution.

To confirm our solution, we substitute x = 2 back into the original equation:

3(2) + 5 = 2(2) + 7

6 + 5 = 4 + 7

11 = 11

Since the equation holds true, our solution x = 2 is correct.

Understanding the Underlying Principles

The process above demonstrates two crucial algebraic principles:

• The Addition/Subtraction Property of Equality: You can add or subtract the same number from both sides of an equation without changing its equality.
• The Multiplication/Division Property of Equality: You can multiply or divide both sides of an equation by the same non-zero number without changing its equality. This property is particularly useful when dealing with equations involving fractions or coefficients other than 1.

Extending the Concepts: More Complex Equations

The techniques used to solve 3x + 5 = 2x + 7 can be extended to solve more complex linear equations. Consider these examples:

• Equations with Fractions: For equations containing fractions, the first step is often to eliminate the fractions by multiplying both sides by the least common denominator (LCD) of the fractions.

• Equations with Parentheses: If the equation contains parentheses, you must first use the distributive property to expand the parentheses before proceeding with the steps outlined above.

• Equations with Multiple Variables: While this specific example only involves one variable, the principles remain the same when solving equations with multiple variables. You would use similar algebraic manipulations to isolate one variable in terms of the others.

Practical Applications and Real-World Examples

Linear equations like this are fundamental to many real-world applications across various fields:

• Physics: Solving for unknown variables in equations describing motion, forces, and energy.
• Engineering: Calculating dimensions, forces, and material properties in structural design.
• Economics: Modeling relationships between variables like supply, demand, and price.
• Finance: Calculating interest, loan repayments, and investment growth.
• Computer Science: Developing algorithms and solving problems in programming.

Practice Problems

To reinforce your understanding, try solving these similar equations:

1. 4x + 3 = x + 9
2. 2x - 5 = 7x + 10
3. (x/2) + 4 = 6
4. 3(x + 2) = 15
5. 5x + 2 = 3x -8

Advanced Techniques and Further Exploration

For more complex equations beyond linear equations, you may need to explore more advanced techniques such as:

• Quadratic Equations: Equations of the form ax² + bx + c = 0. These require methods like factoring, completing the square, or using the quadratic formula.
• Simultaneous Equations: Systems of equations with multiple variables that need to be solved simultaneously. Methods include substitution, elimination, and matrix methods.
• Exponential and Logarithmic Equations: Equations involving exponential and logarithmic functions. These require specific techniques involving logarithms and exponential properties.

Conclusion: Mastering the Fundamentals

Solving the equation 3x + 5 = 2x + 7, though seemingly straightforward, provides a solid foundation for understanding fundamental algebraic principles. Mastering these principles opens the door to solving more complex equations and tackling various real-world problems across different disciplines. Practice is key—the more you work through problems, the more confident and proficient you'll become in manipulating equations and finding solutions. Remember to always verify your solutions by substituting them back into the original equation to ensure accuracy. This iterative approach ensures a strong grasp of algebraic concepts and their practical applications. Through consistent practice and a solid understanding of the underlying principles, you can confidently tackle more intricate mathematical challenges.