Solving 6x - 3 = -51 A Step By Step Guide
Finding the solution to algebraic equations is a fundamental skill in mathematics. In this article, we will delve into the process of solving the equation 6x - 3 = -51. We will break down each step, providing a clear and concise explanation to help you understand the underlying principles. By the end of this guide, you will not only know the answer but also grasp the methodology for solving similar equations. This is an essential skill for anyone studying algebra or related fields, as it forms the basis for more complex problem-solving techniques. Let's begin by understanding the basics of algebraic equations and how to isolate the variable.
Understanding Algebraic Equations
Before diving into the specifics of the equation 6x - 3 = -51, it's crucial to understand the basic structure of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions can contain variables (usually represented by letters like x, y, or z), constants (numbers), and mathematical operations (addition, subtraction, multiplication, and division). The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. In simpler terms, we want to find the value of 'x' that, when plugged into the equation, makes the left side equal to the right side. This involves isolating the variable on one side of the equation, which is achieved by performing the same operations on both sides to maintain the balance. Think of an equation as a weighing scale; to keep it balanced, any change on one side must be mirrored on the other.
The key principles to remember when solving equations are:
- Maintaining Balance: Whatever operation you perform on one side of the equation, you must perform the same operation on the other side.
- Inverse Operations: Use inverse operations to isolate the variable. For example, the inverse of addition is subtraction, and the inverse of multiplication is division.
- Order of Operations: While simplifying expressions within the equation, remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Understanding these principles is crucial for tackling any algebraic equation, including the one we're about to solve. Now, let's move on to the first step in solving 6x - 3 = -51: isolating the term containing the variable.
Step 1: Isolate the Term with the Variable
In the equation 6x - 3 = -51, our primary goal is to isolate the term containing the variable, which is 6x. To do this, we need to eliminate the constant term, which is -3, from the left side of the equation. The inverse operation of subtraction is addition, so we will add 3 to both sides of the equation. This is a critical step, as it maintains the balance of the equation, ensuring that the equality remains true.
Adding 3 to both sides, we get:
6x - 3 + 3 = -51 + 3
Simplifying both sides, we have:
6x = -48
By adding 3 to both sides, we have successfully isolated the term 6x on the left side of the equation. This brings us one step closer to finding the value of 'x'. The next step involves isolating the variable 'x' itself. To do this, we will use the inverse operation of multiplication, which is division. We will divide both sides of the equation by the coefficient of 'x', which is 6. This will give us the value of 'x' and the solution to the equation. Understanding this step is crucial for solving linear equations, as it demonstrates how to isolate a variable multiplied by a constant. Now, let's proceed to the next step and divide both sides by 6.
Step 2: Isolate the Variable
Now that we have the equation 6x = -48, the next step is to isolate the variable 'x'. In this case, 'x' is being multiplied by 6. To isolate 'x', we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by 6. Remember, it's essential to perform the same operation on both sides to maintain the balance of the equation. This is a fundamental principle in solving algebraic equations and ensures that the solution remains accurate.
Dividing both sides by 6, we get:
(6x) / 6 = -48 / 6
Simplifying both sides, we have:
x = -8
By dividing both sides by 6, we have successfully isolated the variable 'x' and found its value. This value, x = -8, is the solution to the equation 6x - 3 = -51. This means that if we substitute -8 for 'x' in the original equation, the equation will hold true. To ensure our solution is correct, it's always a good practice to verify it by plugging the value back into the original equation. This is the final step in the problem-solving process and provides confidence in the accuracy of the answer. Let's move on to the next section to verify our solution.
Step 3: Verify the Solution
To ensure that x = -8 is indeed the correct solution to the equation 6x - 3 = -51, we need to substitute -8 for 'x' in the original equation and check if the equation holds true. This process is called verification and is a crucial step in problem-solving. It helps us catch any potential errors made during the solution process. By verifying the solution, we can confidently say that we have solved the equation correctly.
Substituting x = -8 into the original equation, we get:
6(-8) - 3 = -51
Now, let's simplify the left side of the equation:
-48 - 3 = -51
-51 = -51
As we can see, the left side of the equation is equal to the right side, which means that our solution x = -8 is correct. Verification is a powerful tool in mathematics, as it provides a way to double-check our work and ensure accuracy. It's a practice that should be adopted whenever solving equations or any mathematical problem. Now that we have verified our solution, we can confidently state the final answer.
Final Answer: x = -8
After carefully solving the equation 6x - 3 = -51 and verifying our solution, we have determined that the value of 'x' is -8. This means that when we substitute -8 for 'x' in the original equation, the equation holds true. We arrived at this solution by following a step-by-step process:
- Isolating the term with the variable: We added 3 to both sides of the equation to eliminate the constant term on the left side.
- Isolating the variable: We divided both sides of the equation by 6 to isolate 'x'.
- Verifying the solution: We substituted -8 for 'x' in the original equation to confirm that the equation holds true.
This problem demonstrates the fundamental principles of solving algebraic equations, including the importance of maintaining balance and using inverse operations. By understanding these principles, you can confidently tackle a wide range of algebraic problems. The ability to solve equations is a crucial skill in mathematics and has applications in various fields, including science, engineering, and economics. Therefore, mastering this skill is essential for academic and professional success. We hope this guide has provided a clear and concise explanation of the solution process. Remember to practice regularly to solidify your understanding and improve your problem-solving skills.
Therefore, the final answer is B. -8.
