Solving The Equation (6x-5)+(6x+5)=1 A Step-by-Step Guide
Understanding the Problem
In this article, we will delve into the step-by-step solution of the algebraic equation (6x-5)+(6x+5)=1. This equation falls under the category of linear equations, which are fundamental in mathematics and appear in various applications across science and engineering. Our primary goal is to isolate the variable x to determine its value that satisfies the given equation. Linear equations, characterized by having variables raised to the power of one, are the simplest type of algebraic expressions to solve. Mastering the techniques to solve such equations is crucial for more advanced mathematical concepts. Understanding the properties of equality, such as the addition, subtraction, multiplication, and division properties, is essential for manipulating equations effectively. These properties allow us to perform the same operation on both sides of the equation without changing its solution set. In the process of solving (6x-5)+(6x+5)=1, we will use the distributive property to simplify expressions, combine like terms, and apply inverse operations to isolate x. This equation provides a clear pathway to demonstrate the fundamental algebraic principles involved in solving linear equations, making it an excellent exercise for students and anyone looking to refresh their algebra skills. Before diving into the step-by-step solution, it's important to recognize the structure of the equation and the role each term plays in the overall balance. The parentheses indicate expressions that need to be simplified or combined, and the equal sign signifies that the expressions on both sides have the same value. By methodically applying algebraic principles, we can efficiently find the value of x that makes the equation true. The solution we derive will not only solve the specific equation but also illustrate a general approach applicable to a wide range of linear equations. Understanding and practicing these methods are essential for building a solid foundation in algebra.
Step-by-Step Solution
To solve the equation (6x-5)+(6x+5)=1, we will follow a series of algebraic steps to isolate the variable x. This involves simplifying the equation, combining like terms, and applying inverse operations to both sides of the equation. By methodically following these steps, we can arrive at the solution for x. Our initial focus will be on removing the parentheses and simplifying each side of the equation. This simplification is a crucial step in making the equation more manageable and revealing the underlying structure. We will then combine like terms, which involves grouping together terms with the same variable and constants. This step reduces the number of terms in the equation, making it easier to solve. After combining like terms, we will apply inverse operations to both sides of the equation to isolate x. Inverse operations are operations that undo each other; for example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. By carefully applying these operations, we can systematically eliminate terms and coefficients until we are left with x alone on one side of the equation. The final step is to state the solution, which is the value of x that satisfies the original equation. This value can be verified by substituting it back into the original equation and checking that both sides are equal. Through this step-by-step process, we will demonstrate a clear and logical approach to solving linear equations, providing a model that can be applied to other similar problems. Understanding each step and the reasoning behind it is crucial for developing strong problem-solving skills in algebra. Let's begin by removing the parentheses and simplifying the equation.
Step 1: Remove Parentheses
The first step in solving the equation (6x-5)+(6x+5)=1 is to remove the parentheses. In this case, the parentheses are simply grouping terms, and there is no coefficient or negative sign multiplying the expressions inside. Therefore, we can remove the parentheses directly without changing the terms. Removing parentheses is a fundamental step in simplifying algebraic expressions, as it allows us to combine like terms more easily. When dealing with parentheses, it's important to consider the order of operations (PEMDAS/BODMAS), which dictates that we should perform operations inside parentheses first. However, in this particular equation, there are no operations to perform within the parentheses, so we can proceed directly to their removal. This step simplifies the visual complexity of the equation, making it easier to identify and combine like terms. By removing the parentheses, we transition from a more complex expression to a simpler one that highlights the underlying structure of the equation. This process of simplification is crucial in algebra, as it transforms complex problems into more manageable forms. After removing the parentheses, the equation becomes 6x - 5 + 6x + 5 = 1, which is now ready for the next step of combining like terms. This initial simplification sets the stage for the subsequent steps, leading us closer to the solution for x. The ability to efficiently remove parentheses is a key skill in algebraic manipulation, allowing for more straightforward application of other algebraic principles. Now, let's proceed to combining like terms to further simplify the equation.
Step 2: Combine Like Terms
After removing the parentheses, our equation is now 6x - 5 + 6x + 5 = 1. The next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power, or constant terms. In this equation, we have two terms with the variable x (6x and 6x) and two constant terms (-5 and +5). Combining like terms involves adding or subtracting the coefficients of the terms with the same variable and adding or subtracting the constant terms. This process simplifies the equation by reducing the number of terms, making it easier to isolate the variable x. Combining like terms is a fundamental skill in algebra, and it relies on the distributive property and the commutative property of addition. By grouping and combining like terms, we are essentially simplifying the equation without changing its fundamental meaning. The process of combining like terms helps to reveal the underlying structure of the equation and makes it easier to see the relationship between the variable and the constants. In this case, the terms 6x and 6x are combined to give 12x, and the constant terms -5 and +5 cancel each other out, resulting in 0. This simplification significantly reduces the complexity of the equation, bringing us closer to isolating the variable x. By combining like terms, we are streamlining the equation and preparing it for the final steps of solving for x. This step is crucial in many algebraic problems, as it often transforms complex expressions into simpler, more manageable forms. Let's proceed with the simplified equation and solve for x.
Step 3: Simplify the Equation
After combining like terms in the equation 6x - 5 + 6x + 5 = 1, we arrived at a simplified form. The terms 6x and 6x combined to give 12x, and the constants -5 and +5 canceled each other out, resulting in 0. Therefore, the simplified equation is 12x = 1. This simplified equation is much easier to solve than the original equation, and it clearly shows the relationship between the variable x and the constant term. Simplifying equations is a critical step in solving algebraic problems, as it reduces complexity and makes the solution more apparent. By simplifying, we are essentially stripping away the extraneous terms and focusing on the core relationship between the variable and the constants. The ability to simplify equations efficiently is a key skill in algebra, and it relies on a solid understanding of algebraic principles and properties. In this case, the simplification process has transformed the equation into a straightforward form where isolating the variable x is a simple task. The simplified equation 12x = 1 is a classic example of a linear equation in one variable, where the goal is to find the value of x that makes the equation true. This equation can be solved by applying the inverse operation of multiplication, which is division. By dividing both sides of the equation by the coefficient of x, we can isolate x and find its value. The simplified form of the equation highlights the direct relationship between x and the constant term, making the final step of solving for x relatively straightforward. Let's proceed with the final step of isolating x.
Step 4: Isolate x
Now that we have the simplified equation 12x = 1, the final step is to isolate x. To do this, we need to undo the multiplication by 12. The inverse operation of multiplication is division, so we will divide both sides of the equation by 12. Dividing both sides of an equation by the same non-zero number maintains the equality, which is a fundamental principle in algebra. This principle allows us to manipulate equations without changing their solutions. Isolating the variable is the ultimate goal in solving algebraic equations, as it gives us the value of the variable that satisfies the equation. The process of isolation involves applying inverse operations to eliminate all other terms and coefficients surrounding the variable. In this case, dividing both sides of the equation 12x = 1 by 12 will isolate x on the left side. When we divide 12x by 12, we get x. When we divide 1 by 12, we get 1/12. Therefore, the solution to the equation is x = 1/12. This result means that the value of x that makes the equation (6x-5)+(6x+5)=1 true is 1/12. The process of isolating x demonstrates the power of inverse operations in solving algebraic equations. By systematically applying inverse operations, we can unravel the equation and find the value of the variable. This skill is essential for solving a wide range of algebraic problems, from simple linear equations to more complex equations and systems of equations. Now that we have found the value of x, it's a good practice to verify our solution by substituting it back into the original equation. This verification step ensures that our solution is correct and that we have not made any errors in our calculations. Let's proceed with verifying the solution.
Verification of the Solution
To ensure the accuracy of our solution, we need to verify that the value x = 1/12 satisfies the original equation (6x-5)+(6x+5)=1. Verification is a crucial step in the problem-solving process, as it helps to identify any potential errors made during the solution process. This step involves substituting the value of x back into the original equation and checking if both sides of the equation are equal. If the left-hand side (LHS) equals the right-hand side (RHS) after the substitution, then our solution is correct. Verification not only confirms the correctness of the solution but also reinforces our understanding of the equation and the algebraic principles involved. It is a good practice to verify the solution for every algebraic problem, especially in tests and exams, to avoid losing marks due to careless errors. The verification process also provides an opportunity to double-check our calculations and reasoning, ensuring that we have followed the correct steps. In this case, we will substitute x = 1/12 into the original equation (6x-5)+(6x+5)=1 and simplify both sides. The goal is to show that the LHS and RHS are equal, which will confirm that x = 1/12 is indeed the correct solution. This verification step is a powerful tool for building confidence in our problem-solving abilities and ensuring the accuracy of our results. Let's proceed with the substitution and simplification to verify the solution.
Step 1: Substitute x = 1/12 into the Original Equation
The first step in verifying the solution x = 1/12 is to substitute this value into the original equation (6x-5)+(6x+5)=1. This substitution will replace the variable x with its numerical value, allowing us to simplify the equation and check if both sides are equal. Substitution is a fundamental technique in algebra, used to evaluate expressions and solve equations. It involves replacing a variable with a specific value and then performing the necessary calculations. In this case, we are substituting x = 1/12 into the equation to determine if this value satisfies the equation. The substitution process transforms the algebraic equation into an arithmetic expression, which can then be simplified using the order of operations. Careful substitution is crucial to ensure accuracy in the verification process. Any error in substitution can lead to an incorrect verification result. After substituting x = 1/12, the equation becomes (6(1/12)-5)+(6(1/12)+5)=1**. This equation now contains only numerical values, and we can proceed to simplify it. The substituted equation highlights the importance of accurate calculations and attention to detail in the verification process. Each term in the equation must be evaluated correctly to determine if the equality holds true. By carefully substituting and simplifying, we can confidently verify the correctness of our solution. Now, let's proceed to simplify the equation after the substitution.
Step 2: Simplify the Equation after Substitution
After substituting x = 1/12 into the original equation, we have (6(1/12)-5)+(6(1/12)+5)=1**. The next step is to simplify this equation by performing the arithmetic operations. This involves multiplying, adding, and subtracting the terms within the parentheses and then combining the results. Simplification is a crucial process in verifying the solution, as it allows us to determine if the left-hand side (LHS) of the equation equals the right-hand side (RHS). The order of operations (PEMDAS/BODMAS) must be followed to ensure accurate simplification. This means performing multiplication before addition and subtraction. In this case, we first multiply 6 by 1/12 in both parentheses. This gives us 1/2 in both cases. The equation then becomes (1/2-5)+(1/2+5)=1. Next, we perform the operations inside the parentheses. 1/2 - 5 equals -9/2, and 1/2 + 5 equals 11/2. So the equation becomes (-9/2)+(11/2)=1. Now, we add the two fractions: -9/2 + 11/2 = 2/2, which simplifies to 1. Therefore, the LHS of the equation simplifies to 1, which is equal to the RHS of the equation. This confirms that our solution x = 1/12 is correct. The simplification process demonstrates the importance of following the correct order of operations and performing accurate calculations. By carefully simplifying the equation, we have successfully verified that our solution satisfies the original equation. This verification step provides confidence in our solution and reinforces our understanding of the algebraic principles involved. Now, let's state the final solution.
Final Answer
After solving the equation (6x-5)+(6x+5)=1 and verifying the solution, we have determined that the value of x that satisfies the equation is 1/12. This final answer is the result of a systematic process that involved removing parentheses, combining like terms, simplifying the equation, isolating the variable, and verifying the solution. Each step in this process was crucial in arriving at the correct answer. The solution x = 1/12 represents the point where the equation holds true, meaning that when this value is substituted into the original equation, both sides of the equation are equal. Finding the final answer is the culmination of the problem-solving process, and it demonstrates our ability to apply algebraic principles and techniques to solve equations. The final answer is not just a numerical value; it is the solution to the problem, and it provides valuable information about the relationship between the variables and constants in the equation. In this case, the final answer x = 1/12 provides a specific value that makes the equation balance. Stating the final answer clearly and concisely is an important part of the problem-solving process. It communicates the result of our work and provides a clear conclusion to the problem. The process of finding and verifying the final answer reinforces our understanding of algebraic concepts and strengthens our problem-solving skills. The solution x = 1/12 is a precise and accurate answer to the given equation, and it reflects a thorough and methodical approach to problem-solving. Therefore, the final answer to the equation (6x-5)+(6x+5)=1 is x = 1/12. This concludes the step-by-step solution and verification of the equation.
