Solving Linear Equations A Step-by-Step Guide

Introduction to Linear Equations

In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of algebraic problem-solving. These equations, characterized by a variable raised to the power of one, appear frequently in various scientific and real-world applications. Mastering the technique to solve them efficiently is crucial for anyone venturing into mathematics or related fields. This guide delves into a specific instance of a linear equation, demonstrating a systematic approach to isolate the variable and find its value.

The Anatomy of a Linear Equation

Before we plunge into the solution, it's important to understand the basic structure of a linear equation. Typically, it features one or more variables combined with constants and coefficients, all linked by mathematical operations such as addition, subtraction, multiplication, and division. Our target equation for today's discussion is 7x + 7.96 = 5x + 11. Here, 'x' represents the unknown variable whose value we aim to determine. The coefficients 7 and 5 are the numbers multiplied by the variable x, while 7.96 and 11 are constants. The equation symbolizes a balance, implying that the expression on the left side of the equals sign has the same value as the expression on the right side. The key to solving for x is to manipulate the equation while maintaining this balance, ultimately isolating x on one side.

The Guiding Principles of Solving Equations

The beauty of mathematics lies in its consistency. When solving equations, we adhere to principles that ensure the integrity of our operations. The golden rule is that any operation performed on one side of the equation must be mirrored on the other side. This maintains the balance and ensures that the value of x we obtain is accurate. For instance, if we subtract a number from the left side, we must subtract the same number from the right side. Similarly, if we multiply one side by a constant, we must multiply the other side by the same constant. These operations are performed strategically to simplify the equation and bring us closer to isolating the variable. There are also various strategies for combining like terms, distributing, and simplifying expressions that should be understood before tackling more complicated equations. It is always essential to double-check one’s work and substitute the obtained value of x back into the original equation to verify the solution. This practice helps prevent errors and reinforces comprehension of the solving process.

Step-by-Step Solution of 7x + 7.96 = 5x + 11

Let's tackle the equation 7x + 7.96 = 5x + 11 head-on. Our mission is to find the value of 'x' that satisfies this equation. We'll proceed meticulously, step by step, employing algebraic manipulations to isolate 'x'.

Step 1: Grouping Like Terms

The first maneuver involves grouping like terms. Our objective here is to bring all terms containing 'x' to one side of the equation and all constant terms to the other side. To achieve this, we'll subtract 5x from both sides of the equation. This action cancels out the 5x term on the right side and consolidates the 'x' terms on the left side. The equation now transforms to 7x - 5x + 7.96 = 11. The next move is to address the constant term. To isolate the 'x' terms further, we subtract 7.96 from both sides. This action eliminates the constant term on the left side, shifting it to the right side. Our equation now reads 7x - 5x = 11 - 7.96. These manipulations are critical in simplifying the equation and paving the way for isolating the variable.

Step 2: Simplifying the Equation

With like terms grouped, our next task is to simplify the equation. On the left side, we combine 7x and -5x, which yields 2x. On the right side, we subtract 7.96 from 11, resulting in 3.04. The simplified equation now stands as 2x = 3.04. This simplification is a crucial step as it reduces the complexity of the equation and brings us closer to isolating 'x'. By combining similar terms, we’ve made the equation more manageable and easier to solve. This principle of simplification is ubiquitous in mathematics and is an indispensable tool in solving any kind of equation, not just linear ones.

Step 3: Isolating the Variable

Now comes the pivotal step: isolating the variable 'x'. To achieve this, we need to undo the operation that's currently linked to 'x'. In our case, 'x' is multiplied by 2. The inverse operation of multiplication is division. So, we'll divide both sides of the equation by 2. This action will effectively free 'x' from the coefficient 2 on the left side. The equation transforms into x = 3.04 / 2. Performing the division, we find that x = 1.52. This is the solution to our equation. The value of x that satisfies the equation 7x + 7.96 = 5x + 11 is 1.52. It’s important to note that the strategy of using inverse operations to isolate a variable is a core technique in algebra and forms the basis for solving more intricate equations as well.

Verification of the Solution

Before declaring victory, it's prudent to verify our solution. This step ensures that the value we've calculated for 'x' indeed satisfies the original equation. To verify, we substitute x = 1.52 back into the original equation 7x + 7.96 = 5x + 11. Substituting, we get 7(1.52) + 7.96 = 5(1.52) + 11. Performing the multiplications, we have 10.64 + 7.96 = 7.6 + 11. Adding the terms on both sides, we find 18.6 = 18.6. This equality confirms that our solution is correct. The left side of the equation equals the right side, validating that x = 1.52 is the true solution. This verification process is a cornerstone of accurate problem-solving in mathematics, preventing errors and reinforcing understanding of the equation-solving process.

Conclusion

In this guide, we navigated through the process of solving the linear equation 7x + 7.96 = 5x + 11. We dissected the equation, grouped like terms, simplified, isolated the variable, and verified our solution. Through these steps, we discovered that x = 1.52 is the value that satisfies the equation. This methodical approach is universally applicable to solving linear equations. The key takeaways are the importance of grouping like terms, using inverse operations, and verifying the solution. Mastering these skills not only allows you to solve linear equations efficiently but also forms a strong foundation for more advanced mathematical concepts. Keep practicing, and you'll find solving equations becomes second nature.