Solving Linear Equations A Step-by-Step Guide For 4x + 13 = 3x + 15

Jul 12, 2025 by ADMIN 68 views

Introduction

In mathematics, solving equations is a fundamental skill. Equations represent relationships between variables and constants, and finding the solution means determining the value(s) of the variable(s) that make the equation true. This article will focus on solving a linear equation with one variable, which is a common type of problem encountered in algebra. The specific equation we will address is 4x + 13 = 3x + 15. This equation involves a single variable, 'x', and our goal is to isolate 'x' on one side of the equation to find its value. Understanding how to solve such equations is crucial for various applications in mathematics, science, engineering, and everyday problem-solving. Linear equations form the basis for more complex mathematical concepts, making it essential to master the techniques involved in solving them. This article will provide a step-by-step solution to the equation, explaining the underlying principles and methods used. We will also discuss the importance of verifying the solution to ensure its correctness. By the end of this article, you will have a clear understanding of how to solve similar linear equations and appreciate the significance of this skill in mathematics. Whether you are a student learning algebra or someone looking to refresh your mathematical knowledge, this guide will provide you with the necessary tools and insights to confidently tackle such problems.

Problem Statement

The equation we aim to solve is 4x + 13 = 3x + 15. This is a linear equation because the highest power of the variable 'x' is 1. The equation states that the expression on the left-hand side (4x + 13) is equal to the expression on the right-hand side (3x + 15). Our objective is to find the value of 'x' that satisfies this equality. To achieve this, we will employ algebraic manipulations to isolate 'x' on one side of the equation. This involves performing operations that maintain the balance of the equation, such as adding or subtracting the same quantity from both sides, or multiplying or dividing both sides by the same non-zero quantity. The process of solving this equation demonstrates the fundamental principles of algebraic manipulation and equation solving. It highlights the importance of maintaining equality while simplifying the equation to isolate the variable of interest. This type of problem is a cornerstone of algebra and serves as a building block for more advanced mathematical concepts. Understanding how to solve linear equations is essential for tackling a wide range of mathematical problems and real-world applications. By carefully following the steps outlined in the solution, you will gain a solid understanding of the techniques involved and be able to apply them to similar problems.

Solution

To solve the equation 4x + 13 = 3x + 15, we will follow a step-by-step approach that involves isolating the variable 'x' on one side of the equation. This process involves performing algebraic operations that maintain the equality of the equation. Each step is carefully chosen to simplify the equation and bring us closer to the solution. First, we need to gather the terms containing 'x' on one side and the constant terms on the other side. This is achieved by adding or subtracting terms from both sides of the equation. The key principle here is that whatever operation is performed on one side of the equation must also be performed on the other side to maintain the balance. This ensures that the solution remains valid. By systematically applying these steps, we can reduce the equation to a simple form where the value of 'x' is easily determined. The solution process demonstrates the power of algebraic manipulation in solving equations and highlights the importance of following a logical and consistent approach. Once we have found a potential solution, it is crucial to verify it by substituting it back into the original equation. This step ensures that the solution is correct and that no errors were made during the solution process. The following steps will guide you through the solution of the equation 4x + 13 = 3x + 15.

Step 1: Subtract 3x from both sides

Our initial goal is to consolidate the 'x' terms on one side of the equation. To do this, we subtract 3x from both sides of the equation: 4x + 13 - 3x = 3x + 15 - 3x. This step is crucial because it begins the process of isolating 'x'. By subtracting 3x from both sides, we maintain the equality of the equation while moving the 'x' term from the right side to the left side. This manipulation simplifies the equation and brings us closer to finding the value of 'x'. The result of this subtraction is a new equation that is equivalent to the original equation but is easier to work with. This step illustrates the fundamental principle of maintaining balance in an equation: whatever operation is performed on one side must also be performed on the other side. Understanding this principle is essential for solving equations correctly. The subtraction of 3x from both sides is a strategic move that sets the stage for further simplification and ultimately leads to the solution of the equation. This step is a clear example of how algebraic manipulation can be used to transform an equation into a more manageable form.

After performing the subtraction, the equation becomes:

x + 13 = 15

Step 2: Subtract 13 from both sides

Now that we have simplified the equation to x + 13 = 15, our next step is to isolate 'x' completely. To do this, we need to eliminate the constant term (+13) from the left side of the equation. We achieve this by subtracting 13 from both sides: x + 13 - 13 = 15 - 13. This step is a direct application of the principle of maintaining equality in an equation. By subtracting 13 from both sides, we ensure that the equation remains balanced while effectively removing the constant term from the left side. This manipulation brings us one step closer to finding the value of 'x'. The subtraction of 13 is a strategic move that directly targets the isolation of the variable. It demonstrates the power of algebraic operations in simplifying equations and revealing the solution. This step is crucial because it transforms the equation into a simple form where the value of 'x' can be easily determined. By carefully performing this subtraction, we are setting the stage for the final step in the solution process. This step highlights the importance of understanding the properties of equality and how they can be used to solve equations.

After subtracting 13 from both sides, the equation simplifies to:

x = 2

Verification

To ensure the correctness of our solution, it is essential to verify that x = 2 indeed satisfies the original equation. This involves substituting the value of x back into the original equation and checking if the left-hand side (LHS) is equal to the right-hand side (RHS). This process is a crucial step in problem-solving as it helps to identify any errors that may have occurred during the solution process. Verification provides confidence in the solution and confirms that the algebraic manipulations were performed correctly. It also reinforces the understanding of the equation and the relationship between the variable and the constants. The verification step is not just a formality; it is an integral part of the solution process. It demonstrates a thorough approach to problem-solving and highlights the importance of accuracy in mathematics. By performing this step, we can be certain that our solution is valid and that we have correctly solved the equation. The verification process also serves as a learning opportunity, allowing us to review the steps taken and identify any potential areas for improvement. The following steps will guide you through the verification process for our solution.

Substitute x = 2 into the original equation

Our original equation is 4x + 13 = 3x + 15. To verify our solution, x = 2, we substitute this value into the equation: 4(2) + 13 = 3(2) + 15. This substitution is a direct application of the definition of a solution: a value that makes the equation true when substituted for the variable. By replacing 'x' with 2, we are transforming the equation into a numerical statement that we can evaluate. This step is crucial because it allows us to check if the left-hand side (LHS) and the right-hand side (RHS) of the equation are equal when x = 2. The substitution is a straightforward process, but it requires careful attention to ensure that the value is correctly placed in the equation. This step is a key part of the verification process and sets the stage for the evaluation of the numerical statement. By performing this substitution, we are taking the first step towards confirming the validity of our solution. This step highlights the importance of understanding the meaning of a solution and how it relates to the original equation.

Evaluate both sides of the equation

After substituting x = 2 into the original equation, we have 4(2) + 13 = 3(2) + 15. Now, we need to evaluate both the left-hand side (LHS) and the right-hand side (RHS) of the equation separately. This involves performing the arithmetic operations in the correct order (multiplication before addition). Evaluating both sides will give us two numerical values that we can compare. If the LHS is equal to the RHS, then our solution, x = 2, is correct. If they are not equal, it indicates that there was an error in our solution process, and we need to re-examine our steps. The evaluation step is a critical part of the verification process because it provides concrete evidence of whether our solution satisfies the equation. It demonstrates the importance of accuracy in arithmetic and algebraic manipulations. By carefully evaluating both sides, we can confidently determine the validity of our solution. This step reinforces the understanding of the equation and the relationship between the variable and the constants. The evaluation process is a clear demonstration of how mathematical operations can be used to verify the correctness of a solution.

Left-hand side (LHS): 4(2) + 13 = 8 + 13 = 21
Right-hand side (RHS): 3(2) + 15 = 6 + 15 = 21

Since LHS = RHS (21 = 21), our solution x = 2 is correct.

Conclusion

In conclusion, we have successfully solved the equation 4x + 13 = 3x + 15 and found that the solution is x = 2. We achieved this by following a step-by-step approach that involved isolating the variable 'x' on one side of the equation through algebraic manipulations. These manipulations included subtracting terms from both sides of the equation to maintain equality and simplify the expression. The key principle throughout the solution process was to perform the same operations on both sides of the equation to ensure that the balance was maintained. After finding the solution, we verified its correctness by substituting x = 2 back into the original equation. This verification step confirmed that our solution indeed satisfies the equation, providing confidence in our result. The process of solving this linear equation demonstrates the fundamental principles of algebra and highlights the importance of systematic problem-solving. Understanding how to solve such equations is crucial for various applications in mathematics and other fields. This article has provided a clear and concise guide to solving linear equations, and the techniques discussed can be applied to a wide range of similar problems. By mastering these techniques, you will be well-equipped to tackle more complex mathematical challenges.