Solving 8x + 7 - 6(x + 1) = 5x + 3 A Step-by-Step Guide
Introduction
In this comprehensive guide, we will delve into the process of solving the algebraic equation 8x + 7 - 6(x + 1) = 5x + 3. Equations like this are fundamental in mathematics and appear in various fields, including physics, engineering, and economics. Mastering the techniques to solve them is crucial for anyone pursuing these disciplines. Our goal is to provide a clear, step-by-step solution, ensuring that even those with a basic understanding of algebra can follow along. We will break down each step, explain the underlying principles, and offer insights into why these methods work. By the end of this guide, you will not only be able to solve this particular equation but also gain a deeper understanding of algebraic problem-solving strategies applicable to a wide range of similar problems. This includes understanding the importance of the order of operations, the distributive property, combining like terms, and isolating the variable. We will also highlight common mistakes to avoid and offer tips for checking your answers. Whether you are a student brushing up on your algebra skills or someone looking to expand your mathematical knowledge, this guide is designed to help you succeed.
Understanding the Basics of Algebraic Equations
Before we jump into solving the equation, it's essential to grasp the foundational concepts of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions are typically composed of variables (represented by letters like 'x'), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division). The primary goal when solving an equation is to find the value(s) of the variable(s) that make the equation true. This involves manipulating the equation using various algebraic principles until the variable is isolated on one side, and its value is revealed on the other. For example, in the equation 8x + 7 - 6(x + 1) = 5x + 3, our aim is to determine the value of 'x' that satisfies this equality. This process often involves simplifying the expressions on both sides of the equation, combining like terms, and applying inverse operations to isolate the variable. A key principle in solving equations is maintaining balance. Whatever operation you perform on one side of the equation, you must also perform on the other side to preserve the equality. Understanding these fundamental concepts is crucial for tackling more complex equations and algebraic problems. In the following sections, we will apply these principles step-by-step to solve the given equation, providing a clear and methodical approach to problem-solving.
Step 1: Distribute the -6
The first step in solving the equation 8x + 7 - 6(x + 1) = 5x + 3 is to simplify the expression by distributing the -6 across the terms inside the parentheses. The distributive property states that a(b + c) = ab + ac. Applying this to our equation, we multiply -6 by both 'x' and '1'. This gives us -6 * x = -6x and -6 * 1 = -6. Substituting these results back into the equation, we get: 8x + 7 - 6x - 6 = 5x + 3. This step is crucial because it eliminates the parentheses, making the equation easier to manipulate. Distributing correctly ensures that we account for the multiplication across all terms within the parentheses, which is a common area for errors. By carefully applying the distributive property, we can simplify the equation and prepare it for further steps. This step demonstrates the importance of following the order of operations (PEMDAS/BODMAS), which dictates that we perform operations within parentheses before addition and subtraction. Mastering the distributive property is essential for solving various algebraic equations and is a fundamental skill in algebra. In the next step, we will further simplify the equation by combining like terms on the left side.
Step 2: Combine Like Terms on the Left Side
After distributing the -6, our equation now reads 8x + 7 - 6x - 6 = 5x + 3. The next step is to combine like terms on the left side of the equation. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with 'x' (8x and -6x) and two constant terms (7 and -6). Combining 8x and -6x, we add their coefficients (8 and -6) to get 2x. Combining the constant terms 7 and -6, we add them to get 1. Replacing these combined terms back into the equation, we get: 2x + 1 = 5x + 3. This step simplifies the equation by reducing the number of terms, making it easier to isolate the variable 'x'. Combining like terms is a fundamental algebraic technique that helps to organize and simplify expressions. It is crucial to ensure that you are only combining terms that have the same variable and exponent. This process reduces the complexity of the equation, bringing us closer to finding the value of 'x'. By simplifying one side of the equation, we make the subsequent steps, such as isolating the variable, more manageable. In the next step, we will move the terms with 'x' to one side of the equation and the constants to the other side.
Step 3: Move Terms with 'x' to One Side
Now that our equation is simplified to 2x + 1 = 5x + 3, we need to move all the terms containing 'x' to one side of the equation. A common strategy is to move the term with the smaller coefficient of 'x' to the side with the larger coefficient to avoid dealing with negative coefficients unnecessarily. In this case, we have 2x on the left side and 5x on the right side. To move the 2x term to the right side, we subtract 2x from both sides of the equation. This maintains the balance of the equation and isolates the 'x' terms on one side. Subtracting 2x from both sides gives us: 2x + 1 - 2x = 5x + 3 - 2x. Simplifying this, we get 1 = 3x + 3. This step is crucial because it begins the process of isolating the variable 'x'. By moving all the 'x' terms to one side, we are one step closer to finding the value of 'x'. It is important to perform the same operation on both sides of the equation to maintain equality. This step demonstrates the principle of inverse operations, where we use the opposite operation (subtraction in this case) to move terms across the equals sign. In the next step, we will isolate the constant terms on the other side of the equation.
Step 4: Isolate the Constant Terms
After moving the 'x' terms, our equation is now 1 = 3x + 3. The next step is to isolate the constant terms on one side of the equation. We currently have the constant term '3' on the right side, along with the 'x' term. To isolate the constant terms, we need to move this '3' to the left side of the equation. We do this by subtracting 3 from both sides of the equation, maintaining the balance. Subtracting 3 from both sides gives us: 1 - 3 = 3x + 3 - 3. Simplifying this, we get -2 = 3x. This step is a critical part of the process of solving for 'x' as it further isolates the variable. By performing the same operation on both sides, we ensure that the equation remains balanced. This step, like the previous one, utilizes the principle of inverse operations to move terms across the equals sign. Isolating the constant terms prepares us for the final step, where we will solve for 'x' by dividing both sides of the equation by the coefficient of 'x'. In the next step, we will complete the solution by isolating 'x' and finding its value.
Step 5: Solve for x
Our equation is now simplified to -2 = 3x. The final step in solving for 'x' is to isolate the variable by dividing both sides of the equation by the coefficient of 'x', which is 3 in this case. Dividing both sides by 3 gives us: -2 / 3 = 3x / 3. Simplifying this, we get x = -2/3. This is the solution to the equation 8x + 7 - 6(x + 1) = 5x + 3. We have successfully isolated 'x' and found its value. This step demonstrates the final application of inverse operations, using division to undo the multiplication by 3. It is essential to perform this step accurately to arrive at the correct solution. Once we have found the value of 'x', it is always a good practice to check our answer by substituting it back into the original equation to ensure that it satisfies the equality. This helps to verify that we have not made any errors in our calculations. In the next section, we will discuss how to check our solution and confirm its correctness.
Checking the Solution
After solving for 'x', it's crucial to verify our solution to ensure accuracy. Our solution is x = -2/3. To check this, we substitute this value back into the original equation: 8x + 7 - 6(x + 1) = 5x + 3. Replacing 'x' with '-2/3', we get: 8(-2/3) + 7 - 6(-2/3 + 1) = 5(-2/3) + 3. Now, we simplify each side of the equation separately. On the left side, we have: 8*(-2/3) = -16/3. Then, -2/3 + 1 = -2/3 + 3/3 = 1/3. So, -6 * (1/3) = -2. Therefore, the left side becomes: -16/3 + 7 - 2 = -16/3 + 5 = -16/3 + 15/3 = -1/3. On the right side, we have: 5*(-2/3) = -10/3. So, the right side becomes: -10/3 + 3 = -10/3 + 9/3 = -1/3. Since both sides of the equation equal -1/3, our solution x = -2/3 is correct. This step is essential because it catches any potential errors made during the solving process. By substituting the solution back into the original equation and verifying that both sides are equal, we gain confidence in our answer. Checking the solution is a fundamental practice in mathematics and is highly recommended for all algebraic problems.
Common Mistakes to Avoid
When solving equations like 8x + 7 - 6(x + 1) = 5x + 3, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them. One frequent mistake is incorrectly distributing the negative sign. When distributing -6 across (x + 1), it's crucial to remember to multiply both 'x' and '1' by -6. Forgetting to apply the negative sign to both terms can lead to errors. Another common mistake is combining unlike terms. Only terms with the same variable and exponent can be combined. For example, 8x and -6x can be combined, but 8x and 7 cannot. Mixing up these terms can lead to an incorrect simplification of the equation. Another error occurs when not performing the same operation on both sides of the equation. Maintaining balance is essential in solving equations. Any operation performed on one side must be performed on the other side to preserve equality. Failing to do so will result in an incorrect solution. Finally, arithmetic errors can occur during the simplification process. Careless mistakes in addition, subtraction, multiplication, or division can lead to an incorrect value for 'x'. To avoid these mistakes, it's helpful to double-check each step, show your work clearly, and take your time. By being mindful of these common errors, you can increase your accuracy and confidence in solving algebraic equations.
Conclusion
In conclusion, solving the equation 8x + 7 - 6(x + 1) = 5x + 3 involves a series of steps that require careful application of algebraic principles. We began by distributing the -6 across the terms inside the parentheses, then combined like terms on the left side of the equation. Next, we moved the 'x' terms to one side and the constant terms to the other side, isolating the variable 'x'. Finally, we solved for 'x' by dividing both sides of the equation by the coefficient of 'x', arriving at the solution x = -2/3. We then emphasized the importance of checking the solution by substituting it back into the original equation to verify its correctness. This step is crucial for catching any potential errors and ensuring accuracy. We also discussed common mistakes to avoid, such as incorrect distribution of the negative sign, combining unlike terms, not performing the same operation on both sides, and arithmetic errors. By understanding these potential pitfalls, you can improve your problem-solving skills and avoid making these mistakes. Mastering the techniques for solving algebraic equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. By following a systematic approach and paying attention to detail, you can confidently tackle similar problems and enhance your mathematical proficiency.
