Solving 3x - 5 = 16 For X A Step By Step Guide

In the realm of mathematics, solving equations is a fundamental skill. This article aims to provide a comprehensive, step-by-step guide on how to solve the equation 3x - 5 = 16 for x. We will break down the process into manageable steps, ensuring that you not only arrive at the correct answer but also understand the underlying principles involved. Whether you're a student grappling with algebra or simply looking to brush up on your math skills, this guide will offer clarity and confidence.

Understanding the Basics of Algebraic Equations

Before diving into the specific solution, it's crucial to grasp the basic concepts of algebraic equations. An equation is a mathematical statement that asserts the equality of two expressions. In our case, the equation 3x - 5 = 16 states that the expression on the left side (3x - 5) is equal to the expression on the right side (16). The goal of solving an equation is to isolate the variable (in this case, x) on one side of the equation, thereby determining its value. This involves performing operations on both sides of the equation to maintain the equality while simplifying the expression. These operations typically include addition, subtraction, multiplication, and division. The key principle is that whatever operation you perform on one side of the equation, you must perform the same operation on the other side to preserve the balance. This ensures that the equality remains valid throughout the solving process. Understanding this principle is fundamental to mastering algebraic equations and will be crucial in tackling more complex problems in the future. Remember, each step taken should aim to simplify the equation and bring you closer to isolating the variable, which will ultimately reveal its value.

Step 1: Isolating the Term with 'x'

The initial step in solving the equation 3x - 5 = 16 involves isolating the term containing the variable x. In this instance, the term is 3x. To achieve this, we need to eliminate the constant term, which is -5, from the left side of the equation. The principle of maintaining equality dictates that any operation performed on one side must also be performed on the other. Therefore, to eliminate -5, we perform the inverse operation, which is adding 5. By adding 5 to both sides of the equation, we effectively cancel out the -5 on the left side. This operation transforms the equation as follows:

3x - 5 + 5 = 16 + 5

This simplifies to:

3x = 21

This step is crucial as it brings us closer to isolating x and determining its value. We have successfully removed the constant term from the side containing the variable, making the next step of isolating x much more straightforward. Remember, this process of isolating terms is a fundamental technique in solving algebraic equations, and mastering it will enable you to tackle a wide range of mathematical problems. The ability to strategically manipulate equations by performing the same operation on both sides is the cornerstone of algebraic problem-solving.

Step 2: Solving for 'x'

Now that we have the equation 3x = 21, the next step is to isolate x completely. The term 3x represents 3 multiplied by x. To undo this multiplication and isolate x, we need to perform the inverse operation, which is division. We will divide both sides of the equation by 3. This maintains the equality and allows us to solve for x. Dividing both sides by 3, we get:

(3x) / 3 = 21 / 3

This simplifies to:

x = 7

Therefore, the solution to the equation 3x - 5 = 16 is x = 7. This value of x satisfies the original equation, meaning that when you substitute 7 for x in the equation, both sides will be equal. This step demonstrates the power of inverse operations in solving equations. By systematically applying the appropriate inverse operations, we can unravel the equation and reveal the value of the unknown variable. This process is a cornerstone of algebra and is applicable to a wide range of equation-solving scenarios. The ability to identify the necessary inverse operation and apply it correctly is a key skill for anyone studying mathematics.

Verification: Ensuring the Solution is Correct

To ensure that our solution x = 7 is correct, it's always a good practice to verify it by substituting the value back into the original equation. This step helps to catch any potential errors made during the solving process and confirms the accuracy of the solution. Substituting x = 7 into the original equation 3x - 5 = 16, we get:

3(7) - 5 = 16

Now, we simplify the left side of the equation:

21 - 5 = 16

16 = 16

Since the left side of the equation equals the right side (16 = 16), our solution x = 7 is indeed correct. This verification process highlights the importance of checking your work in mathematics. It not only confirms the accuracy of the solution but also reinforces the understanding of the equation and the steps taken to solve it. By verifying our solution, we can have confidence in our answer and the problem-solving process. This practice is particularly valuable in more complex mathematical problems, where errors can be easily made. The habit of verifying solutions ensures accuracy and builds a solid foundation for further mathematical studies.

Conclusion: Mastering Linear Equations

In conclusion, we have successfully solved the equation 3x - 5 = 16 for x, arriving at the solution x = 7. This process involved isolating the term with x, performing inverse operations, and verifying the solution to ensure its accuracy. Solving linear equations like this is a fundamental skill in algebra and forms the basis for more advanced mathematical concepts. By understanding the principles of equation solving, such as maintaining equality and using inverse operations, you can confidently tackle a wide range of mathematical problems. Remember, practice is key to mastering these skills. The more equations you solve, the more comfortable and proficient you will become. This guide has provided a clear, step-by-step approach to solving this specific equation, but the principles discussed are applicable to many other types of equations as well. With a solid understanding of these fundamentals, you'll be well-equipped to tackle more complex mathematical challenges in the future. Keep practicing, keep learning, and you'll continue to grow your mathematical abilities.

This article demonstrates a clear and concise method for solving a basic linear equation. The steps outlined, including isolating the variable and performing inverse operations, are fundamental concepts in algebra. By mastering these techniques, students and anyone interested in mathematics can confidently approach and solve a wide variety of equations.