In this comprehensive guide, we will delve into the process of solving the algebraic equation 2(2x - 5) = 3(x + 2). This type of equation, a linear equation in one variable, is fundamental in algebra and appears frequently in various mathematical contexts. Understanding how to solve these equations is crucial for building a strong foundation in mathematics. We will break down the solution into clear, manageable steps, ensuring a thorough understanding of each operation involved. Our discussion will encompass the underlying principles of algebraic manipulation, emphasizing the importance of maintaining equality throughout the solution process. By meticulously examining each step, we aim to not only find the solution but also to illuminate the reasoning behind the method, enhancing your problem-solving skills in algebra and beyond.
Step-by-Step Solution
1. Distribute the constants on both sides of the equation.
Our initial step in solving the equation 2(2x - 5) = 3(x + 2) involves applying the distributive property. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms within a set of parentheses. In essence, it states that a(b + c) = ab + ac. This property is crucial for simplifying expressions and equations, as it enables us to remove parentheses and combine like terms. Let's apply this property to both sides of our equation.
On the left side, we have 2(2x - 5). Applying the distributive property, we multiply 2 by both terms inside the parentheses: 2 * 2x and 2 * -5. This gives us 4x - 10. The distributive property is not merely a mechanical rule; it's rooted in the basic principles of arithmetic and the way we understand multiplication as repeated addition. When we multiply 2 by the expression (2x - 5), we are essentially adding the expression (2x - 5) to itself. This expansion allows us to deal with individual terms and to simplify the expression for further manipulation.
Similarly, on the right side of the equation, we have 3(x + 2). Again, we apply the distributive property, multiplying 3 by both terms inside the parentheses: 3 * x and 3 * 2. This results in 3x + 6. The application of the distributive property here follows the same logic as on the left side, ensuring that we account for the multiplication of the constant by each term within the parentheses. Understanding this principle is key to accurately expanding and simplifying algebraic expressions. The distributive property serves as a bridge, connecting the concise form of an expression with parentheses to its expanded form, which is often easier to manipulate in solving equations.
After applying the distributive property to both sides, our equation now looks like this: 4x - 10 = 3x + 6. This transformation is a significant step forward, as it eliminates the parentheses, making the equation more accessible for the subsequent steps of solving for x. By understanding and correctly applying the distributive property, we lay the groundwork for a smoother and more accurate solution process.
2. Move the terms with 'x' to one side of the equation.
After distributing the constants, the equation is now in the form 4x - 10 = 3x + 6. The next crucial step in isolating 'x' involves rearranging the terms so that all terms containing 'x' are on one side of the equation. This process relies on the fundamental principle of algebraic manipulation: maintaining equality. Whatever operation we perform on one side of the equation, we must perform the same operation on the other side to ensure the equation remains balanced.
In this case, we aim to move the term '3x' from the right side to the left side. To achieve this, we subtract '3x' from both sides of the equation. Subtracting '3x' from both sides is akin to removing the same quantity from both sides of a balance scale, thus preserving the balance. This operation is justified by the properties of equality, which allow us to add or subtract the same value from both sides without altering the equation's solution set. Mathematically, this can be represented as follows: (4x - 10) - 3x = (3x + 6) - 3x.
Performing the subtraction, we simplify the left side of the equation: 4x - 3x - 10. Combining the 'x' terms, we get 1x, which is simply written as 'x'. So, the left side becomes x - 10. On the right side of the equation, the '3x' terms cancel each other out: 3x - 3x = 0. This leaves us with just the constant term, 6. Therefore, the right side simplifies to 6. The act of canceling out terms is a direct result of the additive inverse property, which states that adding a number to its negative results in zero. This property is instrumental in simplifying algebraic expressions and isolating variables.
After this step, the equation is significantly simplified to x - 10 = 6. By strategically subtracting '3x' from both sides, we have successfully moved all the terms containing 'x' to the left side, paving the way for the final step of isolating 'x'. This manipulation highlights the power of algebraic techniques in transforming equations into more manageable forms. The beauty of algebra lies in its ability to systematically unravel the complexities of an equation, bringing us closer to the solution with each step.
3. Isolate 'x' by adding 10 to both sides.
Having successfully moved the 'x' terms to one side, our equation now stands as x - 10 = 6. The final step in solving for 'x' involves isolating 'x' completely on one side of the equation. This means we need to eliminate the constant term, '-10', from the left side. To do this, we employ the same principle of maintaining equality that guided our previous steps: we perform the same operation on both sides of the equation.
In this case, we add 10 to both sides of the equation. Adding 10 to both sides is the inverse operation of subtracting 10, and it allows us to cancel out the '-10' term on the left side. This is another application of the additive inverse property, where adding a number to its negative results in zero. The concept of inverse operations is fundamental in algebra, as it provides a systematic way to undo operations and isolate the variable we are trying to solve for. Mathematically, we represent this operation as follows: (x - 10) + 10 = 6 + 10.
On the left side of the equation, adding 10 to x - 10 results in the '-10' and '+10' canceling each other out: -10 + 10 = 0. This leaves us with just 'x' on the left side, which is exactly what we aimed to achieve. On the right side of the equation, we perform the addition: 6 + 10. This simple arithmetic operation yields a result of 16. Therefore, the right side of the equation simplifies to 16. The act of adding constants is a basic arithmetic operation, but its significance in algebra cannot be overstated. It allows us to shift terms across the equality sign, bringing us closer to the solution with each step.
After this step, the equation is elegantly simplified to x = 16. By adding 10 to both sides, we have successfully isolated 'x', revealing its value. This final equation provides a clear and concise solution to our original problem. The journey from the initial equation to this solution highlights the power of algebraic manipulation, demonstrating how a series of logical steps can unravel the complexities of an equation and reveal its hidden solution. The solution x = 16 is not just a number; it's the culmination of a process, a testament to the systematic nature of algebra and its ability to solve problems.
Solution
Therefore, the solution to the equation 2(2x - 5) = 3(x + 2) is x = 16. This solution represents the value of 'x' that makes the equation true. We can verify this solution by substituting x = 16 back into the original equation and checking if both sides are equal. This process of verification is a crucial step in problem-solving, as it ensures the accuracy of our solution and provides a sense of confidence in our work. The concept of a solution in algebra is fundamental; it's the value that satisfies the equation, making the equation a true statement. The solution is not just a number, but a specific value that balances the equation, making the left side equal to the right side.
Verification
To verify our solution, we substitute x = 16 back into the original equation: 2(2(16) - 5) = 3(16 + 2). This substitution replaces the variable 'x' with its numerical value, allowing us to evaluate both sides of the equation. Verification is not merely a formality; it's an essential step in the problem-solving process. It acts as a safeguard, ensuring that our solution is correct and that no errors were made during the algebraic manipulation. This step reinforces the understanding that a solution is not just a random number, but a value that specifically satisfies the given equation.
First, let's simplify the left side of the equation. We start by performing the operation inside the parentheses: 2 * 16 = 32. So, the expression inside the parentheses becomes 32 - 5. Then, we perform the subtraction: 32 - 5 = 27. Now, we multiply this result by the constant outside the parentheses: 2 * 27 = 54. Therefore, the left side of the equation simplifies to 54. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial in this simplification process. Following the correct order ensures that we perform the operations in the appropriate sequence, leading to an accurate result.
Next, we simplify the right side of the equation. We start by performing the operation inside the parentheses: 16 + 2 = 18. Then, we multiply this result by the constant outside the parentheses: 3 * 18. Performing this multiplication, we get 54. Therefore, the right side of the equation also simplifies to 54. The process of simplifying the right side mirrors the process used for the left side, emphasizing the importance of consistent application of mathematical principles.
Comparing the simplified expressions, we see that both sides of the equation are equal: 54 = 54. This equality confirms that our solution, x = 16, is indeed correct. The verification process provides a satisfying conclusion to the problem-solving journey. It not only confirms the accuracy of our solution but also reinforces our understanding of the equation and the principles of algebra. This process of substitution and simplification solidifies the concept of a solution as a value that satisfies the equation, making both sides equal.
Conclusion
In conclusion, we have successfully solved the equation 2(2x - 5) = 3(x + 2) and found the solution to be x = 16. This journey involved applying fundamental algebraic principles such as the distributive property, maintaining equality by performing the same operations on both sides, and using inverse operations to isolate the variable. The process of solving this equation highlights the systematic nature of algebra, where a series of logical steps leads us to the solution. The solution x = 16 is not just a numerical answer; it represents a specific value that satisfies the equation, making the left side equal to the right side. This understanding of what a solution means is crucial for building a strong foundation in mathematics.
We also emphasized the importance of verification, substituting our solution back into the original equation to confirm its accuracy. This step reinforces the understanding that a solution is not just a guess but a value that truly makes the equation a true statement. The verification process provides a sense of confidence in our work and reinforces the importance of accuracy in mathematical problem-solving. The ability to verify a solution is a valuable skill, as it allows us to catch errors and ensure the correctness of our answers.
Solving linear equations like this is a foundational skill in algebra, and the techniques we have discussed here are applicable to a wide range of mathematical problems. By understanding the underlying principles and practicing these techniques, you can develop your problem-solving abilities and tackle more complex algebraic challenges. The journey of solving an equation is a journey of discovery, where each step brings us closer to the solution and deepens our understanding of mathematical concepts. The satisfaction of finding a solution and verifying its accuracy is a testament to the power and beauty of mathematics. This process not only enhances our problem-solving skills but also cultivates a logical and analytical mindset, which is valuable in various aspects of life. The principles and techniques learned in solving linear equations serve as a building block for more advanced mathematical concepts, paving the way for further exploration and discovery in the world of mathematics.
