Solving 99.8x - 79.3 = 202.7 A Step-by-Step Guide

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In the realm of mathematics, algebraic expressions form the bedrock of problem-solving and analytical thinking. They represent relationships between variables and constants, allowing us to model real-world scenarios and find solutions to complex equations. This article delves into the process of solving a specific algebraic expression, providing a comprehensive guide to the steps involved and the underlying mathematical principles. We will focus on the equation 99.8x - 79.3 = 202.7, demonstrating how to isolate the variable 'x' and arrive at an approximate solution, rounded to the nearest hundredth.

Understanding the Basics of Algebraic Equations

Before diving into the solution, it's crucial to grasp the fundamental concepts of algebraic equations. An equation is a mathematical statement that asserts the equality of two expressions. These expressions can involve variables (symbols representing unknown quantities), constants (fixed numerical values), and mathematical operations such as addition, subtraction, multiplication, and division. The primary goal in solving an equation is to determine the value(s) of the variable(s) that make the equation true.

Key Terms and Concepts:

  • Variable: A symbol, usually a letter (e.g., x, y, z), that represents an unknown quantity.
  • Constant: A fixed numerical value (e.g., 2, -5, 3.14).
  • Coefficient: The numerical factor that multiplies a variable (e.g., in the term 3x, 3 is the coefficient).
  • Term: A single number, variable, or the product of numbers and variables (e.g., 5, x, 2y, -3x²).
  • Equation: A statement that two expressions are equal, connected by an equals sign (=).
  • Solution: The value(s) of the variable(s) that make the equation true.

The Golden Rule of Algebra:

The cornerstone of solving algebraic equations is the principle of maintaining balance. Any operation performed on one side of the equation must also be performed on the other side to preserve the equality. This is often referred to as the "Golden Rule of Algebra." Imagine an equation as a balanced scale; if you add or remove weight from one side, you must do the same on the other side to keep the scale balanced. This principle guides every step in the solution process.

Solving the Equation: 99.8x - 79.3 = 202.7

Now, let's tackle the specific equation: 99.8x - 79.3 = 202.7. Our objective is to isolate the variable 'x' on one side of the equation. We'll achieve this by systematically applying inverse operations, ensuring that we maintain the balance of the equation at each step.

Step 1: Isolating the Term with 'x'

The first step is to isolate the term containing 'x' (which is 99.8x) on one side of the equation. To do this, we need to eliminate the constant term (-79.3) on the left side. We can accomplish this by adding 79.3 to both sides of the equation. Remember, the Golden Rule of Algebra dictates that whatever we do to one side, we must do to the other.

99.  8x - 79.3 + 79.3 = 202.7 + 79.3

Simplifying both sides, we get:

99.  8x = 282

Now, the term with 'x' is isolated on the left side, and we have a simpler equation to work with.

Step 2: Isolating 'x'

The next step is to isolate 'x' itself. Currently, 'x' is being multiplied by 99.8. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 99.8:

(99.8x) / 99.8 = 282 / 99.8

Simplifying, we get:

x = 2.8256513026

This gives us the exact value of 'x'. However, the problem asks us to round the solution to the nearest hundredth. Let's proceed to the final step.

Step 3: Rounding to the Nearest Hundredth

To round a number to the nearest hundredth, we look at the digit in the thousandths place (the third digit after the decimal point). If this digit is 5 or greater, we round up the hundredths digit. If it's less than 5, we leave the hundredths digit as it is.

In our case, x = 2.8256513026. The digit in the thousandths place is 5. Therefore, we round up the hundredths digit (2) to 3.

So, the solution rounded to the nearest hundredth is:

x ≈ 2.83

Therefore, the solution to the algebraic expression 99.8x - 79.3 = 202.7, rounded to the nearest hundredth, is approximately 2.83. This means that if we substitute 2.83 for 'x' in the original equation, the two sides will be approximately equal.

Verifying the Solution

It's always a good practice to verify your solution by substituting it back into the original equation. This helps ensure that you haven't made any errors during the solving process.

Let's substitute x = 2.83 into the original equation:

99.  8(2.83) - 79.3 = 202.7

Calculating the left side:

99.  8 * 2.83 = 282.434

282.  434 - 79.3 = 203.134

The result, 203.134, is very close to 202.7. The slight difference is due to the rounding we performed earlier. If we used a more precise value of 'x' (e.g., 2.8256513026), the two sides would be even closer. This verification step confirms that our solution is reasonable and likely correct.

Importance of Precision and Rounding

In many real-world applications of algebra, the solutions we obtain may not be exact whole numbers. They might involve decimals or fractions that extend to many decimal places. In such cases, rounding becomes necessary to present the solution in a more manageable and practical form. However, it's crucial to understand that rounding introduces a degree of approximation. The level of precision required depends on the specific context of the problem. For instance, in engineering calculations, high precision might be essential, while in everyday estimations, rounding to the nearest whole number might suffice. In the context of this problem, rounding to the nearest hundredth provides a reasonable balance between precision and simplicity.

Applications of Algebraic Expressions

Algebraic expressions are not just abstract mathematical concepts; they have a wide range of applications in various fields, including:

  • Science: Modeling physical phenomena, such as motion, energy, and chemical reactions.
  • Engineering: Designing structures, circuits, and systems.
  • Economics: Analyzing financial markets, predicting economic trends.
  • Computer Science: Developing algorithms, writing software.
  • Everyday Life: Calculating budgets, planning trips, solving puzzles.

Understanding how to manipulate and solve algebraic expressions is a fundamental skill that empowers us to tackle problems in these diverse areas.

Conclusion

Solving algebraic expressions is a fundamental skill in mathematics with applications across various disciplines. In this article, we have demonstrated a step-by-step approach to solving the equation 99.8x - 79.3 = 202.7, culminating in the approximate solution x ≈ 2.83, rounded to the nearest hundredth. We emphasized the importance of the Golden Rule of Algebra, the process of isolating the variable, and the significance of rounding in practical applications. By mastering these concepts, you'll be well-equipped to tackle a wide range of algebraic problems and unlock the power of mathematical reasoning.

This problem provides a valuable illustration of how algebraic equations can be used to model and solve real-world problems. By understanding the underlying principles and practicing problem-solving techniques, you can develop your mathematical skills and gain a deeper appreciation for the power of algebra.