Solving Linear Equations A Step-by-Step Guide To 7x + 7.96 = 5x + 11

Introduction

In the realm of mathematics, linear equations form the bedrock of countless problem-solving scenarios. These equations, characterized by their simple yet powerful structure, provide a framework for understanding relationships between variables and constants. Mastering the art of solving linear equations is not merely an academic exercise; it's a crucial skill that permeates various fields, from physics and engineering to economics and computer science. This article delves into the step-by-step process of solving a specific linear equation, $7x + 7.96 = 5x + 11$, while simultaneously illuminating the underlying principles that govern this fundamental mathematical operation.

Before we embark on the journey of solving this equation, it's essential to grasp the core concepts that underpin linear equations. A linear equation, at its heart, is a mathematical statement that asserts the equality between two expressions. These expressions typically involve variables, which represent unknown quantities, and constants, which are fixed numerical values. The goal of solving a linear equation is to isolate the variable on one side of the equation, thereby determining its value. This process often involves a series of algebraic manipulations, guided by the fundamental principle of maintaining equality on both sides of the equation. Each operation performed on one side must be mirrored on the other, ensuring that the balance is preserved and the solution remains accurate. The equation we'll be tackling, $7x + 7.96 = 5x + 11$, exemplifies a classic linear equation with a single variable, 'x'. Our mission is to unravel the value of 'x' that satisfies this equation, and in doing so, we'll uncover the elegance and precision inherent in mathematical problem-solving.

Step 1: Combine Like Terms

Our initial objective in solving the equation $7x + 7.96 = 5x + 11$ is to consolidate the terms that share a common variable. This strategic maneuver simplifies the equation, paving the way for a more streamlined solution. To achieve this, we focus on gathering the 'x' terms on one side of the equation and the constant terms on the other. The guiding principle here is the preservation of equality; any operation performed on one side must be mirrored on the other to maintain the balance of the equation.

In this particular case, we have two terms involving the variable 'x': $7x$ on the left side and $5x$ on the right side. To bring these terms together, we can subtract $5x$ from both sides of the equation. This operation effectively eliminates the $5x$ term from the right side, while simultaneously introducing a corresponding term on the left side. The equation now transforms into $7x - 5x + 7.96 = 5x - 5x + 11$. Simplifying further, we combine the 'x' terms on the left side, resulting in $2x + 7.96 = 11$. This seemingly small step represents a significant stride towards isolating the variable 'x'.

Next, we turn our attention to the constant terms. We have $7.96$ on the left side and $11$ on the right side. To group these constants together, we subtract $7.96$ from both sides of the equation. This action removes the $7.96$ term from the left side, leaving us with the 'x' term isolated on that side. The equation now becomes $2x + 7.96 - 7.96 = 11 - 7.96$. Simplifying this, we arrive at $2x = 3.04$. At this juncture, the equation has been significantly streamlined. We have successfully combined like terms, effectively separating the variable 'x' from the constants. This lays the foundation for the final step, where we will isolate 'x' and unveil its value.

Step 2: Isolate the Variable

With the equation now simplified to $2x = 3.04$, our next crucial step is to isolate the variable 'x'. This means we need to get 'x' by itself on one side of the equation. The current equation shows 'x' being multiplied by 2. To undo this multiplication, we employ the inverse operation: division. Dividing both sides of the equation by 2 will effectively isolate 'x'.

Applying this principle, we divide both sides of the equation $2x = 3.04$ by 2. This gives us $\frac{2x}{2} = \frac{3.04}{2}$. On the left side, the division cancels out the multiplication by 2, leaving us with just 'x'. On the right side, we perform the division, $3.04 \div 2$, which equals 1.52. Thus, the equation simplifies to $x = 1.52$.

This step marks the culmination of our algebraic journey. We have successfully isolated the variable 'x' and determined its value. The solution to the equation $7x + 7.96 = 5x + 11$ is $x = 1.52$. This value is the unique solution that satisfies the original equation. Substituting 1.52 for 'x' in the original equation will make both sides of the equation equal, confirming our solution.

Isolating the variable is a fundamental technique in solving linear equations. It involves using inverse operations to undo any operations that are being applied to the variable. In this case, we used division to undo multiplication. This principle applies universally to solving various types of equations and is a cornerstone of algebraic manipulation.

Step 3: Verify the Solution

Having arrived at a solution, $x = 1.52$, it is imperative to verify its accuracy. This crucial step ensures that the value we've obtained genuinely satisfies the original equation. Verification involves substituting the solution back into the equation and confirming that both sides of the equation evaluate to the same value. This process acts as a safeguard against potential errors made during the algebraic manipulations.

To verify our solution, we substitute $x = 1.52$ into the original equation, $7x + 7.96 = 5x + 11$. This gives us $7(1.52) + 7.96 = 5(1.52) + 11$. Now, we perform the arithmetic operations on both sides of the equation.

On the left side, $7(1.52)$ equals $10.64$. Adding $7.96$ to this result, we get $10.64 + 7.96 = 18.6$. Thus, the left side of the equation evaluates to $18.6$.

On the right side, $5(1.52)$ equals $7.6$. Adding $11$ to this result, we get $7.6 + 11 = 18.6$. Thus, the right side of the equation also evaluates to $18.6$.

Comparing the values of both sides, we observe that the left side ($18.6$) is indeed equal to the right side ($18.6$). This confirms that our solution, $x = 1.52$, is correct. The verification process not only provides assurance in the accuracy of the solution but also reinforces the understanding of the equation's fundamental nature. It demonstrates that the value of 'x' we found makes the two expressions in the equation equivalent.

Conclusion

In this comprehensive guide, we have meticulously dissected the process of solving the linear equation $7x + 7.96 = 5x + 11$. From the initial step of combining like terms to the final verification of the solution, we have traversed the landscape of algebraic manipulation, guided by the principles of equality and inverse operations. The solution we arrived at, $x = 1.52$, represents the value that satisfies the equation, making both sides of the equation equivalent.

The journey of solving linear equations is not merely a mechanical process; it's an exercise in logical reasoning and problem-solving. Each step, from combining like terms to isolating the variable, is a deliberate action aimed at simplifying the equation and revealing the unknown. The verification step serves as a crucial validation, ensuring the accuracy of the solution and reinforcing the understanding of the equation's fundamental properties.

Mastering the art of solving linear equations is a cornerstone of mathematical proficiency. It equips individuals with the tools to tackle a wide range of problems across diverse fields. The principles and techniques elucidated in this article extend beyond the specific equation we addressed, serving as a foundation for solving more complex equations and mathematical challenges. As you continue your mathematical journey, the skills honed in solving linear equations will undoubtedly prove invaluable, empowering you to unravel the intricacies of the mathematical world.