Solving For Y In The Equation 4(y - 15) + 11 = 3(y - 15) + 11

This article provides a step-by-step solution to the equation 4(y - 15) + 11 = 3(y - 15) + 11, determining the value of 'y' that satisfies the equation. We'll break down the algebraic process, making it easy to understand for anyone looking to enhance their equation-solving skills. This guide is designed to help students, educators, and anyone interested in mathematics to grasp the fundamental principles of solving linear equations.

Understanding the Equation

Before diving into the solution, let's first understand the given equation: 4(y - 15) + 11 = 3(y - 15) + 11. This is a linear equation in one variable, 'y'. Our goal is to isolate 'y' on one side of the equation to find its value. To achieve this, we will use the principles of algebraic manipulation, ensuring that we maintain the equality throughout the process. Understanding the structure of the equation is crucial. We have terms involving 'y', constant terms, and parentheses that indicate multiplication. The order of operations (PEMDAS/BODMAS) will guide us in simplifying the equation. We will first address the parentheses, then combine like terms, and finally isolate 'y'. This systematic approach will help us arrive at the correct solution efficiently and accurately. Recognizing the distributive property's role is key, as we will be multiplying the constants outside the parentheses with the terms inside. Each step is designed to bring us closer to isolating 'y', making the solution clear and concise. By carefully following these steps, we can confidently solve for 'y' and verify our answer. The equation presents a balanced relationship, and our manipulations must respect this balance to ensure the integrity of the solution. Understanding this balance is fundamental to solving any algebraic equation.

Step-by-Step Solution

1. Distribute the constants:

The first step in solving the equation 4(y - 15) + 11 = 3(y - 15) + 11 is to apply the distributive property. This means multiplying the constants outside the parentheses with each term inside the parentheses. On the left side of the equation, we have 4 multiplied by (y - 15). This gives us 4 * y and 4 * -15. Similarly, on the right side, we have 3 multiplied by (y - 15), resulting in 3 * y and 3 * -15. Performing these multiplications, we get 4y - 60 on the left side and 3y - 45 on the right side. The equation now looks like this: 4y - 60 + 11 = 3y - 45 + 11. This step is crucial because it removes the parentheses, making it easier to combine like terms and isolate the variable 'y'. The distributive property is a fundamental concept in algebra and is essential for simplifying equations. It allows us to transform complex expressions into more manageable forms. By carefully applying the distributive property, we ensure that each term within the parentheses is correctly accounted for, maintaining the equality of the equation. This meticulous approach sets the stage for the subsequent steps in solving for 'y'. It's important to double-check the multiplications to avoid errors, as any mistake in this step can propagate through the rest of the solution.

2. Combine like terms:

After distributing the constants, the equation becomes 4y - 60 + 11 = 3y - 45 + 11. The next step is to combine the like terms on each side of the equation. Like terms are those that have the same variable raised to the same power (in this case, 'y') or are constants. On the left side, we have -60 and +11, which are both constants. Combining these, we get -60 + 11 = -49. On the right side, we also have constants -45 and +11. Combining these, we get -45 + 11 = -34. The equation now simplifies to 4y - 49 = 3y - 34. This step reduces the complexity of the equation, making it easier to isolate the variable 'y'. Combining like terms is a fundamental algebraic technique that streamlines the equation-solving process. It involves adding or subtracting terms that share a common variable or are constant values. By simplifying both sides of the equation, we create a clearer pathway to solving for the unknown. This step is essential for organizing the equation and making it more manageable. Accurate combination of like terms is critical to maintaining the balance of the equation and ensuring the correctness of the solution. It's a crucial step in the overall strategy of solving linear equations.

3. Isolate the variable 'y':

Now that we have simplified the equation to 4y - 49 = 3y - 34, the next step is to isolate the variable 'y'. This involves moving all terms containing 'y' to one side of the equation and all constant terms to the other side. To do this, we can subtract 3y from both sides of the equation. This will eliminate the 'y' term from the right side and leave us with a single 'y' term on the left side. Subtracting 3y from both sides, we get 4y - 3y - 49 = 3y - 3y - 34, which simplifies to y - 49 = -34. Next, we need to move the constant term, -49, to the right side of the equation. To do this, we add 49 to both sides. This gives us y - 49 + 49 = -34 + 49, which simplifies to y = 15. By isolating the variable 'y', we have found the value that satisfies the equation. This process is a core technique in algebra, allowing us to solve for unknown quantities. The key is to perform the same operations on both sides of the equation, maintaining the balance and ensuring the equality remains true. Each step brings us closer to the solution, ultimately revealing the value of 'y'. This method of isolating variables is fundamental to solving a wide range of algebraic problems. It demonstrates the power of algebraic manipulation in simplifying and solving equations.

Final Answer

After carefully following the steps of distribution, combining like terms, and isolating the variable, we have arrived at the solution: y = 15. This means that the value of 'y' that satisfies the given equation 4(y - 15) + 11 = 3(y - 15) + 11 is 15. To verify this solution, we can substitute y = 15 back into the original equation and check if both sides are equal. Substituting y = 15, we get 4(15 - 15) + 11 = 3(15 - 15) + 11, which simplifies to 4(0) + 11 = 3(0) + 11, and further to 11 = 11. Since both sides of the equation are equal, our solution is correct. The final answer of y = 15 demonstrates the successful application of algebraic principles to solve the equation. This process highlights the importance of each step, from distributing constants to isolating the variable, in arriving at the correct solution. The ability to solve such equations is a fundamental skill in mathematics and is essential for various applications in science, engineering, and other fields. The verification step is also crucial as it provides confidence in the accuracy of the solution.

Therefore, the solution to the equation 4(y - 15) + 11 = 3(y - 15) + 11 is y = 15, which corresponds to option D) 15.