Solving For X In The Equation 5(x-10) = 30 - 15x

Introduction

In the realm of algebraic equations, solving for the unknown variable, often denoted as 'x', is a fundamental skill. This article delves into the process of solving a specific linear equation: 5(x - 10) = 30 - 15x. We will explore the step-by-step methodology, breaking down each operation to ensure a clear understanding. The goal is not just to arrive at the correct answer but to grasp the underlying principles that govern equation solving. This skill is crucial for various applications in mathematics, science, and engineering.

Understanding the Equation

The equation we aim to solve is 5(x - 10) = 30 - 15x. This is a linear equation because the highest power of the variable 'x' is 1. Linear equations are prevalent in numerous real-world scenarios, making their solution a vital skill. The equation consists of two sides, the left-hand side (LHS) and the right-hand side (RHS), separated by an equals sign (=). Our objective is to isolate 'x' on one side of the equation to determine its value. To achieve this, we will employ algebraic manipulations, ensuring that any operation performed on one side is also applied to the other to maintain the equation's balance. The solution set provided includes four potential values for x: x=1, x=4, x=5, and x=8. Through the following steps, we will identify which, if any, of these values satisfy the given equation.

Step 1: Distribute on the Left-Hand Side

The first step in solving the equation involves simplifying the left-hand side (LHS). We achieve this by applying the distributive property, which states that a(b + c) = ab + ac. In our equation, 5(x - 10), 'a' is 5, 'b' is 'x', and 'c' is -10. Applying the distributive property, we multiply 5 by both 'x' and -10. This yields 5 * x = 5x and 5 * -10 = -50. Therefore, the LHS simplifies to 5x - 50. The equation now becomes 5x - 50 = 30 - 15x. This step is crucial because it eliminates the parentheses, making the equation easier to manipulate. By distributing the 5, we have expanded the expression, paving the way for combining like terms and isolating 'x'. Remember, the distributive property is a cornerstone of algebraic manipulation, allowing us to simplify expressions and solve equations more effectively. This initial simplification is essential for the subsequent steps in solving for x.

Step 2: Combine Like Terms

After distributing on the left-hand side, our equation is now 5x - 50 = 30 - 15x. The next step is to combine like terms. This involves gathering all terms containing 'x' on one side of the equation and all constant terms on the other side. To do this, we can add 15x to both sides of the equation. This eliminates the -15x term on the right-hand side and introduces a 15x term on the left-hand side. The equation becomes 5x + 15x - 50 = 30. Next, we add 50 to both sides of the equation to eliminate the -50 term on the left-hand side. This gives us 5x + 15x = 30 + 50. Now, we simplify both sides by combining the 'x' terms and the constant terms. On the left-hand side, 5x + 15x combines to 20x. On the right-hand side, 30 + 50 adds up to 80. Thus, the equation simplifies to 20x = 80. This step is crucial as it consolidates the 'x' terms and constant terms, bringing us closer to isolating 'x'. Combining like terms is a fundamental algebraic technique that simplifies equations and makes them easier to solve.

Step 3: Isolate x

Following the combination of like terms, our equation stands as 20x = 80. The primary goal now is to isolate x, meaning we want to get 'x' by itself on one side of the equation. To achieve this, we need to eliminate the coefficient that is multiplying 'x', which in this case is 20. The inverse operation of multiplication is division, so we divide both sides of the equation by 20. This ensures that the equation remains balanced, as any operation performed on one side must also be performed on the other. Dividing both sides by 20 gives us (20x) / 20 = 80 / 20. On the left-hand side, 20x divided by 20 simplifies to x. On the right-hand side, 80 divided by 20 equals 4. Therefore, the equation simplifies to x = 4. This step is the culmination of our efforts, where we have successfully isolated 'x' and determined its value. Isolating the variable is a critical step in solving equations, and understanding how to do so is essential for mastering algebra.

Step 4: Verify the Solution

Having found a potential solution, x = 4, it is crucial to verify the solution to ensure its accuracy. This involves substituting the value of 'x' back into the original equation, 5(x - 10) = 30 - 15x, and checking if both sides of the equation are equal. Substituting x = 4 into the equation gives us 5(4 - 10) = 30 - 15(4). First, we simplify the expressions within the parentheses: 4 - 10 = -6. So, the left-hand side becomes 5(-6) = -30. Next, we simplify the right-hand side: 15(4) = 60, so the right-hand side becomes 30 - 60 = -30. Comparing both sides, we have -30 = -30, which is a true statement. This confirms that x = 4 is indeed the correct solution to the equation. Verifying the solution is a vital step in the problem-solving process, as it helps to catch any potential errors made during the algebraic manipulations. By substituting the solution back into the original equation, we ensure that our answer is accurate and reliable.

Conclusion

In conclusion, we have successfully solved the equation 5(x - 10) = 30 - 15x by following a systematic approach. We began by distributing on the left-hand side, then combined like terms, isolated 'x', and finally, verified our solution. The correct solution to the equation is x = 4. This process highlights the fundamental principles of algebra, including the distributive property, combining like terms, and performing inverse operations to isolate the variable. These skills are essential for solving a wide range of algebraic equations and are applicable in various mathematical and real-world contexts. By understanding and mastering these techniques, one can confidently tackle more complex equations and problems. The ability to solve for 'x' is a cornerstone of mathematical proficiency, opening doors to advanced concepts and applications in diverse fields. This step-by-step guide serves as a valuable resource for anyone seeking to enhance their algebraic problem-solving skills.