Solving 3.45x - 2.05x = 3.15 A Step By Step Algebra Guide

Introduction to Algebraic Equations

In the realm of mathematics, algebraic equations form the bedrock of problem-solving and analytical thinking. These equations, characterized by the presence of variables and mathematical operations, allow us to represent real-world scenarios in a symbolic form, thereby facilitating their analysis and resolution. At its core, algebra involves manipulating symbols and applying mathematical principles to uncover the values of unknown quantities, offering a powerful toolkit for deciphering the complexities of the world around us.

Algebraic equations serve as mathematical statements that assert the equality between two expressions. These expressions may consist of constants, variables, and a medley of mathematical operations such as addition, subtraction, multiplication, and division. The primary objective in solving algebraic equations is to isolate the variable, thereby determining its value. This process typically involves the strategic application of algebraic principles, including the manipulation of terms, application of inverse operations, and simplification of expressions. By mastering the art of algebraic problem-solving, individuals gain the ability to tackle a wide array of mathematical challenges, spanning from simple arithmetic problems to complex scientific and engineering applications.

Algebraic equations are not confined to the realm of abstract mathematics; rather, they serve as indispensable tools in numerous fields, including physics, engineering, economics, and computer science. In physics, algebraic equations are employed to model the motion of objects, calculate forces, and analyze electrical circuits. Engineers utilize algebraic equations in the design of structures, the optimization of processes, and the analysis of mechanical systems. Economists leverage algebraic equations to model market dynamics, forecast economic trends, and assess the impact of policy interventions. Computer scientists rely on algebraic equations in the development of algorithms, the design of computer graphics, and the analysis of data. Thus, the versatility and ubiquity of algebraic equations underscore their significance in both theoretical and practical contexts.

Understanding the Equation 3.45x - 2.05x = 3.15

At first glance, the equation 3.45x - 2.05x = 3.15 may appear intimidating, but a closer examination reveals its underlying simplicity. This equation embodies a fundamental algebraic relationship, where the variable 'x' represents an unknown quantity that we seek to determine. The equation presents a linear expression on the left-hand side, consisting of two terms, 3.45x and -2.05x, which are combined through subtraction. The right-hand side of the equation represents a constant value, 3.15. The equation essentially asserts that the difference between 3.45 times 'x' and 2.05 times 'x' is equal to 3.15. To solve this equation, we must employ algebraic techniques to isolate the variable 'x' and uncover its numerical value.

Before delving into the solution process, it is essential to recognize the significance of the equation's structure. The equation's linearity implies that the relationship between 'x' and the constants is straightforward and predictable. This linearity simplifies the solution process, allowing us to employ a systematic approach to isolate the variable. Furthermore, the presence of decimal coefficients, 3.45 and 2.05, adds a layer of complexity, requiring careful attention to arithmetic operations and decimal placement. However, with a methodical approach and a firm grasp of algebraic principles, we can confidently tackle the equation and arrive at the correct solution. The equation serves as a microcosm of the broader landscape of algebraic problems, encapsulating the challenges and rewards inherent in mathematical problem-solving.

The equation 3.45x - 2.05x = 3.15 serves as a representative example of the kinds of equations encountered in various mathematical and scientific disciplines. Its simplicity belies its significance, as it embodies the core principles of algebraic manipulation and equation-solving. By mastering the techniques required to solve this equation, individuals develop a foundation for tackling more complex mathematical challenges. The equation also provides a tangible illustration of the power of algebra in representing real-world relationships and solving practical problems. Whether we are calculating distances, determining costs, or modeling physical phenomena, algebraic equations like this one serve as essential tools for understanding and manipulating the world around us.

Step-by-Step Solution

1. Combine Like Terms

The first step in solving the equation 3.45x - 2.05x = 3.15 involves combining like terms. Like terms are terms that contain the same variable raised to the same power. In this equation, both 3.45x and -2.05x are like terms because they both contain the variable 'x' raised to the power of 1. To combine like terms, we simply add or subtract their coefficients. The coefficient of a term is the numerical factor that multiplies the variable. In this case, the coefficients are 3.45 and -2.05. Performing the subtraction, we get:

3. 45x - 2.05x = (3.45 - 2.05)x = 1.40x

Thus, the equation simplifies to:

4. 40x = 3.15

This simplification represents a crucial step in isolating the variable 'x'. By combining like terms, we have reduced the equation to a more manageable form, making it easier to proceed with the solution process. The principle of combining like terms is a fundamental concept in algebra, applicable to a wide range of equations and expressions. It allows us to condense complex expressions into simpler forms, facilitating subsequent algebraic manipulations. The ability to identify and combine like terms is an essential skill for any aspiring mathematician or problem-solver. This step underscores the importance of recognizing patterns and applying fundamental algebraic principles to streamline the solution process. The simplified equation, 1.40x = 3.15, represents a significant milestone in our journey towards solving for 'x', bringing us closer to the final solution.

2. Isolate the Variable

The next step in solving the equation 1.40x = 3.15 involves isolating the variable 'x'. To isolate a variable, we must perform operations on both sides of the equation to undo any operations that are being performed on the variable. In this case, the variable 'x' is being multiplied by 1.40. To undo this multiplication, we must divide both sides of the equation by 1.40. Dividing both sides of the equation by 1.40, we get:

(1. 40x) / 1.40 = 3.15 / 1.40

This simplifies to:

x = 3.15 / 1.40

Performing the division, we get:

x = 2.25

Thus, the value of 'x' that satisfies the equation is 2.25. This step demonstrates the power of inverse operations in isolating variables and solving equations. The principle of performing the same operation on both sides of an equation ensures that the equality is maintained, allowing us to manipulate the equation without altering its solution. Isolating the variable is a fundamental technique in algebra, applicable to a wide variety of equations and problem-solving scenarios. The ability to strategically apply inverse operations is a crucial skill for anyone seeking to master the art of algebraic manipulation. The solution, x = 2.25, represents the culmination of our efforts, providing the answer to the original equation and validating the steps we have taken.

Verification

To ensure the accuracy of our solution, it is crucial to verify that the value we obtained for 'x' indeed satisfies the original equation. This verification process involves substituting the calculated value of 'x' back into the original equation and confirming that the equation holds true. Substituting x = 2.25 into the original equation, 3.45x - 2.05x = 3.15, we get:

3. 45(2.25) - 2.05(2.25) = 3.15

Performing the multiplications, we get:

4. 7625 - 4.6125 = 3.15

Subtracting the terms on the left-hand side, we get:

5. 15 = 3.15

Since the left-hand side of the equation equals the right-hand side, we have verified that our solution, x = 2.25, is correct. This verification step is an essential practice in mathematics, providing confidence in the accuracy of the solution and highlighting any potential errors in the solution process. It also reinforces the understanding of the equation's structure and the relationship between the variable and the constants. Verification serves as a safeguard against mistakes, ensuring that the final answer is reliable and trustworthy. The successful verification of our solution underscores the power of algebraic techniques and the importance of meticulous attention to detail in mathematical problem-solving. The confirmation of our solution provides closure to the problem-solving process, validating our efforts and solidifying our understanding of the underlying mathematical principles.

Conclusion

In summary, we have successfully solved the algebraic equation 3.45x - 2.05x = 3.15 by systematically applying algebraic principles. The solution process involved combining like terms, isolating the variable, and verifying the solution. We began by combining the like terms 3.45x and -2.05x, simplifying the equation to 1.40x = 3.15. Next, we isolated the variable 'x' by dividing both sides of the equation by 1.40, yielding the solution x = 2.25. Finally, we verified our solution by substituting x = 2.25 back into the original equation, confirming that the equation holds true. This comprehensive approach demonstrates the importance of a methodical and rigorous problem-solving strategy in mathematics.

This exercise highlights the fundamental principles of algebra and their application in solving real-world problems. The ability to manipulate algebraic equations is a crucial skill in various fields, including science, engineering, economics, and computer science. By mastering these techniques, individuals gain the confidence and competence to tackle a wide range of mathematical challenges. The systematic approach employed in this solution can be applied to other algebraic equations, regardless of their complexity. The key lies in understanding the underlying principles, applying them strategically, and verifying the results. The solution to this equation serves as a stepping stone towards more advanced mathematical concepts and problem-solving skills.

In conclusion, the solution of the equation 3.45x - 2.05x = 3.15 exemplifies the power and elegance of algebraic methods. The systematic approach, involving combining like terms, isolating the variable, and verifying the solution, underscores the importance of a structured problem-solving strategy. The value obtained, x = 2.25, represents the numerical solution to the equation, providing a concrete answer to the abstract mathematical statement. This exercise not only reinforces the fundamental principles of algebra but also demonstrates the practical utility of mathematical problem-solving in various domains. The journey from the initial equation to the final solution highlights the interconnectedness of mathematical concepts and the importance of perseverance in the face of challenges. The successful completion of this task serves as a testament to the power of algebraic thinking and its capacity to unravel the complexities of the mathematical world.