Solving For X In 8x - 2 - 5x = 8 A Step-by-Step Guide
In this article, we will delve into the process of solving for x in the equation 8x - 2 - 5x = 8. This is a fundamental algebraic problem that combines several key concepts, including combining like terms and isolating the variable. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide will provide a step-by-step explanation to help you master this type of problem.
Understanding the Basics of Algebraic Equations
Before we jump into the solution, let's briefly discuss the basics of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions are connected by an equals sign (=). Our goal in solving an equation is to find the value (or values) of the variable that makes the equation true. In our case, the variable is x, and we want to find the value of x that satisfies the equation 8x - 2 - 5x = 8. To achieve this, we will use the principles of algebraic manipulation, which involve performing the same operations on both sides of the equation to maintain balance and ultimately isolate the variable.
Key principles in solving equations include the addition property of equality (adding the same quantity to both sides), the subtraction property of equality (subtracting the same quantity from both sides), the multiplication property of equality (multiplying both sides by the same quantity), and the division property of equality (dividing both sides by the same quantity). These properties allow us to simplify the equation step by step until we arrive at the solution. Additionally, understanding the order of operations (PEMDAS/BODMAS) is crucial to correctly simplify expressions within the equation. Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) dictate the sequence in which operations should be performed. By keeping these principles in mind, we can approach solving equations with confidence and accuracy. The ability to manipulate algebraic expressions and equations is a cornerstone of mathematical proficiency, essential not only for academic success but also for various real-world applications involving problem-solving and critical thinking.
Step-by-Step Solution to 8x - 2 - 5x = 8
Step 1: Combine Like Terms
The first step in solving the equation 8x - 2 - 5x = 8 is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this equation, 8x and -5x are like terms. We can combine them by adding their coefficients:
8x - 5x = 3x
So, the equation now becomes:
3x - 2 = 8
Combining like terms simplifies the equation, making it easier to work with. This step is based on the distributive property of multiplication over addition and subtraction. By identifying and combining like terms, we reduce the complexity of the equation and move closer to isolating the variable. In our case, combining 8x and -5x into 3x streamlines the equation and sets the stage for subsequent steps. This process is not only efficient but also fundamental in solving a wide range of algebraic equations. Mastery of combining like terms is essential for success in algebra and other mathematical disciplines, as it allows for the simplification of complex expressions and the efficient solving of equations.
Step 2: Isolate the Variable Term
Next, we need to isolate the variable term, which in this case is 3x. To do this, we want to get rid of the constant term on the left side of the equation, which is -2. We can accomplish this by adding 2 to both sides of the equation. This is based on the addition property of equality, which states that adding the same quantity to both sides of an equation does not change the equality:
3x - 2 + 2 = 8 + 2
This simplifies to:
3x = 10
Isolating the variable term is a crucial step in solving algebraic equations. It sets the stage for the final step, which involves isolating the variable itself. By adding 2 to both sides, we effectively canceled out the -2 on the left side, leaving the term with the variable, 3x, by itself. This process ensures that we maintain the balance of the equation while moving closer to our goal of finding the value of x. The addition property of equality is a fundamental principle in algebra, allowing us to manipulate equations in a way that preserves their validity. Without this property, we would not be able to systematically isolate variables and solve for their values. Understanding and applying this principle correctly is essential for mastering algebraic problem-solving.
Step 3: Solve for x
Now that we have 3x = 10, we can solve for x. To isolate x, we need to divide both sides of the equation by the coefficient of x, which is 3. This is based on the division property of equality, which states that dividing both sides of an equation by the same non-zero quantity does not change the equality:
3x / 3 = 10 / 3
This simplifies to:
x = 10/3
So, the solution to the equation 8x - 2 - 5x = 8 is x = 10/3. This fraction can also be expressed as a mixed number (3 1/3) or as a decimal (approximately 3.33).
Solving for x involves isolating the variable completely, which allows us to determine its value. In this case, dividing both sides by 3 effectively “undoes” the multiplication of x by 3, leaving x by itself. The division property of equality is a powerful tool in algebra, enabling us to unravel the relationships between variables and constants within an equation. By performing this operation, we arrive at the solution, which is the value of x that satisfies the original equation. The ability to solve for variables is a fundamental skill in mathematics, with applications across various fields, including science, engineering, and economics. A clear understanding of the division property of equality and its correct application are key to successful problem-solving in algebra and beyond.
Verification of the Solution
To ensure our solution is correct, we can verify it by substituting x = 10/3 back into the original equation:
8(10/3) - 2 - 5(10/3) = 8
First, we perform the multiplications:
(80/3) - 2 - (50/3) = 8
Next, we convert the whole number 2 into a fraction with a denominator of 3:
(80/3) - (6/3) - (50/3) = 8
Now, we combine the fractions:
(80 - 6 - 50) / 3 = 8
24 / 3 = 8
8 = 8
Since the equation holds true, our solution x = 10/3 is correct.
Verifying the solution is an essential step in the problem-solving process. It provides a check that the value obtained for the variable indeed satisfies the original equation. By substituting the solution back into the equation, we can confirm that both sides of the equation are equal, thereby validating our answer. In this case, we meticulously performed the arithmetic operations, ensuring that each step was accurate. The verification process not only confirms the correctness of the solution but also enhances understanding of the equation and the steps taken to solve it. It reinforces the importance of precision and attention to detail in mathematical problem-solving. Furthermore, it builds confidence in one's ability to solve equations correctly. The practice of verification is highly recommended as it minimizes the risk of errors and promotes a deeper comprehension of algebraic principles.
Common Mistakes to Avoid
When solving equations like 8x - 2 - 5x = 8, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy.
One common mistake is incorrectly combining like terms. For example, some students might mistakenly subtract 5x from 2 instead of combining 8x and -5x. Remember to only combine terms that have the same variable raised to the same power.
Another frequent error is not applying operations to both sides of the equation. For instance, if you add 2 to the left side of the equation, you must also add 2 to the right side to maintain the equality. Failing to do so will lead to an incorrect solution.
Mistakes in arithmetic are also common, such as errors in addition, subtraction, multiplication, or division. These errors can occur when simplifying the equation or when performing the final calculation to find the value of x. It's always a good idea to double-check your calculations to minimize these mistakes.
Additionally, forgetting the order of operations (PEMDAS/BODMAS) can lead to errors. Make sure to perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Finally, some students may fail to verify their solution. Verification is a crucial step that helps catch any errors made during the solving process. By substituting the solution back into the original equation, you can confirm whether it satisfies the equation or not.
By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving algebraic equations. Practice and attention to detail are key to mastering this fundamental skill.
Conclusion
In this guide, we have walked through the process of solving for x in the equation 8x - 2 - 5x = 8. We covered the steps of combining like terms, isolating the variable term, and solving for x. We also emphasized the importance of verifying the solution and avoiding common mistakes. By following these steps and understanding the underlying principles, you can confidently solve similar algebraic equations. Remember, practice is key to mastering these skills, so keep working on problems and reinforcing your knowledge.
The ability to solve algebraic equations is a fundamental skill in mathematics with broad applications in various fields. A solid understanding of these concepts not only aids in academic pursuits but also enhances problem-solving abilities in real-world situations. The systematic approach outlined in this guide, from combining like terms to verifying the solution, provides a framework for tackling a wide range of algebraic problems. Continuous practice and a focus on accuracy will further solidify these skills, empowering individuals to approach mathematical challenges with confidence. The journey of mastering algebra is a rewarding one, opening doors to more advanced mathematical concepts and applications. With consistent effort and a clear understanding of the core principles, success in algebra, and beyond, is within reach.
