Solving For X In The Equation 0.7x - 1.4 = -3.5 A Step By Step Guide

In the realm of mathematics, solving equations is a fundamental skill. This article provides a comprehensive guide to solving for x in the equation 0.7x - 1.4 = -3.5. We will explore the step-by-step process, delve into the underlying principles, and offer insights to enhance your understanding of algebraic problem-solving. This equation, a linear equation in one variable, exemplifies a common type of problem encountered in algebra. Mastering the techniques to solve such equations is crucial for progress in more advanced mathematical concepts. This detailed explanation not only provides the solution but also aims to equip you with the knowledge to tackle similar problems with confidence.

Understanding the Equation

Before diving into the solution, let's break down the equation 0.7x - 1.4 = -3.5. This equation states that when 1.4 is subtracted from 0.7 times a certain number (represented by x), the result is -3.5. Our goal is to isolate x on one side of the equation to determine its value. The equation is linear because the highest power of x is 1. Linear equations are the simplest type of algebraic equations and form the basis for more complex algebraic structures. Recognizing the structure of the equation is the first step in devising a solution strategy. The equation involves decimal numbers, which may require careful handling during the solving process. However, the fundamental algebraic principles remain the same regardless of the type of numbers involved. By understanding the structure of the equation, we can apply appropriate algebraic operations to isolate x and find its value.

Step-by-Step Solution

Step 1: Isolate the Term with x

Our first goal is to isolate the term containing x (0.7x) on one side of the equation. To achieve this, we need to eliminate the constant term (-1.4) from the left side. We can do this by adding 1.4 to both sides of the equation. Adding the same value to both sides maintains the equality, a fundamental principle of algebraic manipulation. This step is crucial as it begins the process of isolating x, bringing us closer to the solution. The equation now becomes:

0. 7x - 1.4 + 1.4 = -3.5 + 1.4

Simplifying, we get:

0. 7x = -2.1

Step 2: Solve for x

Now that we have isolated the term 0.7x, we need to isolate x itself. To do this, we divide both sides of the equation by the coefficient of x, which is 0.7. Dividing both sides by the same non-zero value preserves the equality. This step is the final step in isolating x and finding its value. The equation becomes:

0. 7x / 0.7 = -2.1 / 0.7

Simplifying, we find:

x = -3

Therefore, the value of x that satisfies the equation 0.7x - 1.4 = -3.5 is -3. This solution can be verified by substituting -3 for x in the original equation.

Verification

To ensure the accuracy of our solution, it's crucial to verify it. We substitute x = -3 back into the original equation:

0. 7(-3) - 1.4 = -3.5

-2.1 - 1.4 = -3.5

-3.5 = -3.5

Since the equation holds true, our solution x = -3 is correct. Verification is a critical step in problem-solving, as it confirms the accuracy of the solution and helps to identify any potential errors in the process. By substituting the solution back into the original equation, we can ensure that the left-hand side equals the right-hand side, confirming the validity of our answer. This practice reinforces the understanding of equation solving and promotes carefulness in mathematical calculations.

Key Concepts and Principles

1. The Golden Rule of Algebra

The golden rule of algebra states that whatever operation you perform on one side of an equation, you must perform the same operation on the other side to maintain equality. This principle is the foundation of solving algebraic equations. It ensures that the equation remains balanced throughout the solving process. Whether adding, subtracting, multiplying, or dividing, applying the same operation to both sides preserves the relationship between the expressions on either side of the equation. This rule is not just a mechanical step but a fundamental concept that underpins algebraic manipulation. Understanding and applying this rule correctly is essential for solving any algebraic equation.

2. Inverse Operations

Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. In solving equations, we use inverse operations to isolate the variable. To eliminate a term added to the variable, we subtract it from both sides. To eliminate a term multiplying the variable, we divide both sides by it. The strategic use of inverse operations is the key to simplifying equations and isolating the variable of interest. Recognizing the appropriate inverse operation to apply is a critical skill in equation solving. This technique allows us to systematically peel away the layers of the equation until we arrive at the solution.

3. Order of Operations (PEMDAS/BODMAS)

While not directly applicable in this specific equation due to its simplicity, the order of operations (often remembered by the acronyms PEMDAS or BODMAS) is a crucial concept in mathematics. It dictates the order in which operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding and applying the order of operations is essential for correctly simplifying expressions and solving equations, especially those involving multiple operations. Although our equation was straightforward, adhering to the order of operations is a good habit to cultivate for tackling more complex problems. This principle ensures consistency and accuracy in mathematical calculations.

Common Mistakes to Avoid

1. Incorrectly Applying the Golden Rule

A common mistake is to perform an operation on only one side of the equation, violating the golden rule of algebra. For example, adding 1.4 to the left side but not the right side would lead to an incorrect solution. Always remember to apply any operation to both sides to maintain equality. This error stems from a misunderstanding of the fundamental principle of equation solving. Consistent application of the golden rule is crucial for accurate solutions. Double-checking each step to ensure adherence to this rule is a good practice.

2. Arithmetic Errors

Arithmetic errors, such as incorrect addition, subtraction, multiplication, or division, can lead to wrong answers. In our example, a mistake in adding -3.5 and 1.4 or dividing -2.1 by 0.7 would result in an incorrect solution. Careful calculation and double-checking your work are essential to avoid these errors. Even small arithmetic errors can have significant consequences in the final result. Developing strong arithmetic skills and practicing careful calculation techniques are vital for success in algebra.

3. Forgetting the Sign

Forgetting the sign of a number is another common mistake. For instance, incorrectly handling the negative sign in -3.5 or -2.1 can lead to an incorrect answer. Pay close attention to signs throughout the solving process. Negative signs are a frequent source of errors in algebraic manipulations. Careful attention to detail and consistent use of the correct sign are crucial for accurate results. Developing a habit of double-checking the signs at each step can help prevent this type of mistake.

4. Incorrect Order of Operations

Although not directly relevant in this equation, the incorrect application of the order of operations can be a problem in more complex equations. Make sure to follow the correct order (PEMDAS/BODMAS) to avoid errors. While our equation was relatively simple, understanding the order of operations is essential for tackling more challenging problems. Prioritizing operations correctly ensures accurate simplification and solution of complex expressions and equations.

Practice Problems

To solidify your understanding, try solving these similar equations:

1. 5x + 2.5 = -7.5
2. 2x - 3.6 = 1.4
3. 8x + 4.2 = -9.8

Solving these practice problems will help you reinforce the steps and principles discussed in this article. Practice is key to mastering equation solving. By working through various examples, you will develop confidence and proficiency in applying the techniques learned. These problems provide an opportunity to test your understanding and identify areas where further practice may be needed. Consistent practice is the most effective way to improve your skills in algebra.

Conclusion

Solving for x in the equation 0.7x - 1.4 = -3.5 demonstrates the application of fundamental algebraic principles. By understanding the golden rule of algebra, inverse operations, and the importance of careful calculation, you can confidently tackle similar problems. Remember to verify your solutions and practice regularly to enhance your skills. This comprehensive guide aims to provide a solid foundation for understanding and solving linear equations. Mastery of these skills is crucial for further exploration of mathematical concepts. By following the steps outlined and practicing consistently, you can develop proficiency in algebraic problem-solving.