Solving The Equation 0.05(x-210) = 260 A Step-by-Step Guide

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In the realm of mathematics, equations serve as intricate puzzles, each holding a secret value waiting to be discovered. Today, we embark on a journey to decipher the equation 0.05(x-210) = 260, a seemingly complex expression that can be elegantly untangled with the application of fundamental algebraic principles. This article will serve as your comprehensive guide, leading you through each step of the solution process, while also providing insights into the underlying concepts that make this equation, and equations in general, solvable.

Deciphering the Equation: A Step-by-Step Approach

At first glance, the equation 0.05(x-210) = 260 might appear daunting, but fear not! We will systematically break it down, transforming it into a form where the value of 'x' becomes crystal clear. Our strategy involves a series of algebraic manipulations, each designed to isolate 'x' on one side of the equation, revealing its true identity.

Step 1: Divide both sides by 0.05

The initial hurdle is the coefficient 0.05, which obscures the expression within the parentheses. To overcome this, we employ the fundamental principle of equality: what we do to one side of the equation, we must do to the other. By dividing both sides of the equation by 0.05, we effectively neutralize its influence on the left side.

0. 05(x-210) / 0.05 = 260 / 0.05

This simplifies to:

x - 210 = 5200

Step 2: Isolate 'x' by adding 210 to both sides

Now, 'x' is tantalizingly close to being isolated, but the pesky -210 is still clinging to it. To liberate 'x', we again invoke the principle of equality, adding 210 to both sides of the equation.

x - 210 + 210 = 5200 + 210

This elegantly simplifies to:

x = 5410

The Solution Unveiled: x = 5410

With a flourish, we arrive at the solution: x = 5410. This is the value that, when substituted into the original equation, will make the equation a true statement. To ensure the accuracy of our solution, we can perform a quick verification.

Verifying the Solution: A Crucial Step

In the world of equations, verification is paramount. It's our way of double-checking our work, ensuring that the solution we've obtained is indeed the correct one. To verify our solution, we substitute x = 5410 back into the original equation:

0. 05(5410 - 210) = 260

Simplifying the expression within the parentheses:

0. 05(5200) = 260

Performing the multiplication:

260 = 260

The equation holds true! This confirms that our solution, x = 5410, is indeed the correct value.

The Power of Equations: Unlocking Mathematical Mysteries

The equation we've just solved, 0.05(x-210) = 260, is a testament to the power of mathematics to unravel complex relationships. Equations are the language of the universe, describing the intricate interplay of quantities and variables. By mastering the art of solving equations, we gain the ability to decipher these relationships, unlocking the secrets of the world around us.

Applications in the Real World

The beauty of equations lies not only in their elegance but also in their practicality. They are the cornerstones of countless real-world applications, from engineering and physics to economics and finance. Let's consider a few examples:

  • Engineering: Engineers use equations to design structures, calculate stresses and strains, and optimize performance. For instance, equations are used to determine the optimal shape of a bridge, ensuring its stability and load-bearing capacity.
  • Physics: Physicists rely heavily on equations to describe the laws of motion, gravity, electromagnetism, and quantum mechanics. Equations like F = ma (Newton's second law of motion) and E = mc² (Einstein's famous equation relating energy and mass) are fundamental to our understanding of the universe.
  • Economics: Economists use equations to model economic systems, forecast market trends, and analyze the impact of government policies. For example, equations are used to model supply and demand, predict inflation rates, and assess the effectiveness of fiscal policies.
  • Finance: Financial analysts employ equations to calculate investment returns, manage risk, and price financial instruments. Equations are used to determine the present value of future cash flows, calculate loan payments, and assess the risk of various investments.

Building a Solid Foundation: Essential Algebraic Concepts

Solving equations is not just about following a set of steps; it's about understanding the underlying concepts that make these steps valid. A solid foundation in algebra is crucial for mastering the art of equation solving. Let's revisit some of the key concepts that we've utilized in solving the equation 0.05(x-210) = 260:

  • The Principle of Equality: This fundamental principle states that we can perform the same operation on both sides of an equation without changing its validity. This is the cornerstone of equation solving, allowing us to manipulate equations while preserving the equality.
  • Inverse Operations: To isolate a variable, we often use inverse operations. For example, to undo addition, we subtract; to undo multiplication, we divide. In our equation, we used division to undo the multiplication by 0.05 and addition to undo the subtraction of 210.
  • The Distributive Property: This property allows us to simplify expressions involving parentheses. While we didn't explicitly use the distributive property in this particular equation, it's a valuable tool in solving more complex equations.

Mastering the Art of Equation Solving: Practice Makes Perfect

Like any skill, equation solving requires practice. The more equations you solve, the more comfortable you'll become with the process, and the more adept you'll be at recognizing the appropriate strategies to use. Start with simple equations and gradually work your way up to more complex ones. Don't be afraid to make mistakes; mistakes are valuable learning opportunities.

Conclusion: Embracing the World of Equations

In this journey through the equation 0.05(x-210) = 260, we've not only found the solution (x = 5410) but also gained a deeper appreciation for the power and elegance of equations. Equations are not just abstract mathematical expressions; they are the keys to understanding the world around us. By mastering the art of equation solving, we unlock a world of possibilities, enabling us to solve problems, make predictions, and gain insights into the intricate workings of the universe. So, embrace the challenge, practice diligently, and embark on your own equation-solving adventures!

Remember, every equation is a puzzle waiting to be solved, and with the right tools and techniques, you can unlock its secrets.

This equation-solving journey has equipped you with the skills to tackle similar mathematical challenges. Embrace the power of equations, and continue exploring the fascinating world of mathematics!