If 2x-3 What Is The Value Of X

If 2x - 3 = 0, What is the Value of x? A Comprehensive Guide to Solving Linear Equations

Finding the value of 'x' in the equation 2x - 3 = 0 might seem straightforward, but it forms the bedrock of understanding linear equations – a fundamental concept in algebra and numerous applications across various fields. This comprehensive guide will delve into solving this specific equation, exploring the underlying principles and extending the knowledge to tackle more complex linear equations. We'll cover different approaches, emphasizing the importance of each step and offering practical tips to build a solid foundation in algebra.

Understanding the Equation: 2x - 3 = 0

The equation 2x - 3 = 0 is a linear equation because the highest power of the variable 'x' is 1. Linear equations represent a straight line when graphed. The goal is to isolate 'x' – to find the value of 'x' that makes the equation true. In essence, we're seeking the point where the line represented by 2x - 3 intersects the x-axis (where y = 0).

Key Concepts:

• Variable (x): An unknown quantity we aim to determine.
• Coefficient (2): The number multiplying the variable.
• Constant (-3): A fixed numerical value.
• Equation: A statement asserting the equality of two expressions.

Solving the Equation: Step-by-Step Approach

We'll use the principles of balancing equations to solve for 'x'. The core idea is to perform the same operation on both sides of the equation to maintain equality.

Step 1: Add 3 to Both Sides

Our goal is to isolate the term containing 'x'. To eliminate the constant -3, we add its opposite (3) to both sides of the equation:

2x - 3 + 3 = 0 + 3

This simplifies to:

2x = 3

Step 2: Divide Both Sides by 2

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

2x / 2 = 3 / 2

This results in:

x = 3/2 or x = 1.5

Therefore, the value of x that satisfies the equation 2x - 3 = 0 is 1.5 or 3/2.

Verification: Checking Our Solution

It's crucial to verify our solution by substituting x = 1.5 back into the original equation:

2(1.5) - 3 = 3 - 3 = 0

Since the equation holds true, our solution x = 1.5 is correct. This verification step is essential to ensure accuracy and build confidence in our problem-solving skills.

Expanding Our Understanding: Solving More Complex Linear Equations

The principles used to solve 2x - 3 = 0 can be extended to solve more complex linear equations. Let's explore some examples:

Example 1: 5x + 7 = 12

Step 1: Subtract 7 from both sides:

5x + 7 - 7 = 12 - 7

5x = 5

Step 2: Divide both sides by 5:

5x / 5 = 5 / 5

x = 1

Verification: 5(1) + 7 = 12 (Correct)

Example 2: 3x - 8 = 13

Step 1: Add 8 to both sides:

3x - 8 + 8 = 13 + 8

3x = 21

Step 2: Divide both sides by 3:

3x / 3 = 21 / 3

x = 7

Verification: 3(7) - 8 = 13 (Correct)

Example 3: -2x + 5 = 1

Step 1: Subtract 5 from both sides:

-2x + 5 - 5 = 1 - 5

-2x = -4

Step 2: Divide both sides by -2:

-2x / -2 = -4 / -2

x = 2

Verification: -2(2) + 5 = 1 (Correct)

Equations with Fractions and Decimals

Linear equations can also involve fractions and decimals. The solving process remains the same, but careful attention to arithmetic is crucial.

Example 4: (1/2)x + 3 = 7

Step 1: Subtract 3 from both sides:

(1/2)x + 3 - 3 = 7 - 3

(1/2)x = 4

Step 2: Multiply both sides by 2 (the reciprocal of 1/2):

2 * (1/2)x = 4 * 2

x = 8

Verification: (1/2)(8) + 3 = 7 (Correct)

Example 5: 0.5x - 2 = 1.5

Step 1: Add 2 to both sides:

0.5x - 2 + 2 = 1.5 + 2

0.5x = 3.5

Step 2: Divide both sides by 0.5:

0.5x / 0.5 = 3.5 / 0.5

x = 7

Verification: 0.5(7) - 2 = 1.5 (Correct)

Applications of Linear Equations

Linear equations are fundamental to numerous real-world applications across various fields:

• Physics: Calculating velocity, acceleration, and distance.
• Engineering: Designing structures, analyzing circuits, and modeling systems.
• Economics: Analyzing supply and demand, predicting economic growth, and modeling market behavior.
• Finance: Calculating interest, analyzing investments, and forecasting financial trends.
• Computer Science: Developing algorithms, optimizing processes, and solving computational problems.

Understanding and mastering linear equations is crucial for success in these and many other fields.

Further Exploration: Beyond Linear Equations

While this guide focuses on solving simple linear equations, the principles extend to more complex algebraic concepts:

• Quadratic Equations: Equations where the highest power of the variable is 2. These require different solving techniques, such as factoring, the quadratic formula, or completing the square.
• Simultaneous Equations: Systems of two or more equations with multiple variables that need to be solved together. Methods like substitution or elimination are employed.
• Inequalities: Statements involving inequalities (>, <, ≥, ≤). Solving inequalities requires similar steps to solving equations, but with additional considerations regarding the inequality symbols.

By building a strong foundation in solving linear equations, you'll be well-equipped to tackle these more advanced mathematical concepts.

Conclusion: Mastering the Fundamentals

Solving the equation 2x - 3 = 0 is not just about finding the value of 'x'; it's about understanding the fundamental principles of algebra, including equation balancing and the importance of verification. The skills gained from solving simple linear equations are transferable to more complex algebraic problems and have widespread applications across numerous fields. By mastering these fundamental concepts, you'll build a solid base for future mathematical studies and real-world problem-solving. Remember to practice regularly, exploring various equation types and verifying your solutions to build confidence and proficiency. This consistent practice is key to unlocking a deeper understanding of algebra and its significant role in various disciplines.