Solving For Y In -3(y+7) = 2y-26 A Step-by-Step Guide
Introduction
In this comprehensive guide, we will delve into the step-by-step process of solving for the variable y in the equation -3(y+7) = 2y-26. This is a fundamental algebraic problem that requires a clear understanding of the order of operations, distribution, combining like terms, and isolating the variable. Whether you're a student looking to improve your algebra skills or someone brushing up on their math knowledge, this detailed explanation will provide you with the necessary tools to tackle similar equations with confidence. We will explore each step meticulously, ensuring you grasp the underlying concepts and techniques involved in solving for y.
Solving algebraic equations is a crucial skill in mathematics and has numerous applications in various fields, including physics, engineering, economics, and computer science. The ability to manipulate equations and isolate variables allows us to model real-world scenarios, make predictions, and solve complex problems. Therefore, mastering the techniques for solving equations like -3(y+7) = 2y-26 is not only essential for academic success but also for practical applications in everyday life and professional settings. This article aims to break down the process into manageable steps, providing clear explanations and examples to ensure a thorough understanding. By the end of this guide, you will be well-equipped to solve similar algebraic equations and apply these skills to more advanced mathematical concepts.
The equation -3(y+7) = 2y-26 is a linear equation, which means it involves variables raised to the power of one. Solving linear equations typically involves isolating the variable on one side of the equation by performing algebraic operations. These operations must be applied equally to both sides of the equation to maintain balance and ensure the solution remains valid. The main steps in solving this equation include distributing the constant term, combining like terms, and performing addition, subtraction, multiplication, or division to isolate the variable. Each of these steps will be explained in detail in the subsequent sections, providing you with a clear and methodical approach to solving algebraic equations. Let's begin by understanding the first crucial step: distribution.
Step 1: Distribute the -3
The first crucial step in solving for y in the equation -3(y+7) = 2y-26 involves distributing the -3 across the terms inside the parenthesis. Distribution is a fundamental concept in algebra that allows us to simplify expressions by multiplying a term outside the parenthesis with each term inside the parenthesis. In this case, we need to multiply -3 by both y and +7. This process is essential for eliminating the parenthesis and making the equation easier to manipulate. Understanding and correctly applying the distributive property is vital for solving a wide range of algebraic equations, making it a cornerstone of algebraic problem-solving.
The distributive property states that a( b + c ) = ab + ac. Applying this property to our equation, we multiply -3 by y and -3 by +7. When multiplying -3 by y, we get -3y. Next, we multiply -3 by +7, which results in -21. It’s crucial to pay attention to the signs during this process, as a negative number multiplied by a positive number yields a negative number. Therefore, the result of distributing -3 across (y+7) is -3y - 21. This step simplifies the left side of the equation, allowing us to proceed with the next steps in solving for y. Properly executing distribution is a foundational skill in algebra, and mastering it is key to successfully solving more complex equations.
After distributing the -3, our equation now looks like this: -3y - 21 = 2y - 26. This simplified form is much easier to work with, as we have eliminated the parenthesis and can now focus on combining like terms and isolating the variable y. The next step in our process will involve moving all the terms containing y to one side of the equation and all the constant terms to the other side. This will help us to group similar terms together and further simplify the equation. Remember, the goal is to isolate y on one side of the equation so that we can determine its value. By carefully applying algebraic principles and following a step-by-step approach, we can systematically solve for y. Let’s move on to the next step of combining like terms.
Step 2: Combine Like Terms
After distributing the -3, the equation is now -3y - 21 = 2y - 26. The next strategic step in solving for y involves combining like terms. Combining like terms is a fundamental algebraic technique that simplifies equations by grouping terms that contain the same variable or are constants. In our equation, the terms with the variable y are -3y and 2y, and the constants are -21 and -26. To effectively combine these terms, we need to move them to the same side of the equation. This involves performing operations on both sides to maintain the balance of the equation, a critical principle in algebraic manipulations.
The objective here is to gather all y terms on one side and all constant terms on the other side. A common approach is to move the y terms to the left side and the constants to the right side. To do this, we first need to eliminate the 2y term from the right side. We can achieve this by subtracting 2y from both sides of the equation. This ensures that the equation remains balanced and that the value of y remains consistent. When we subtract 2y from both sides, the equation transforms as follows: -3y - 2y - 21 = 2y - 2y - 26. Simplifying this, we get -5y - 21 = -26. Now, all the y terms are on the left side of the equation.
Next, we need to move the constant term, -21, to the right side of the equation. To do this, we add 21 to both sides. Adding the same value to both sides maintains the equation’s balance and helps isolate the variable. The equation becomes -5y - 21 + 21 = -26 + 21. Simplifying this, we have -5y = -5. At this point, we have successfully combined like terms and isolated the y term on one side of the equation. The equation is now in a much simpler form, making it easier to solve for y. The final step will involve isolating y completely by performing one last operation. Let's proceed to the final step of isolating the variable.
Step 3: Isolate y
Having combined like terms, the equation is now simplified to -5y = -5. The final step in solving for y involves isolating the variable. Isolating the variable means getting y by itself on one side of the equation. In this case, y is being multiplied by -5. To isolate y, we need to perform the inverse operation, which is division. We will divide both sides of the equation by -5. This maintains the balance of the equation and allows us to determine the value of y. Understanding the principle of inverse operations is crucial in solving algebraic equations, as it allows us to undo operations and isolate variables effectively.
To isolate y, we divide both sides of the equation -5y = -5 by -5. This gives us (-5y) / -5 = (-5) / -5. When we perform the division on the left side, the -5 in the numerator and the -5 in the denominator cancel each other out, leaving us with just y. On the right side, we have -5 divided by -5, which equals 1 because a negative number divided by a negative number results in a positive number. Therefore, the equation simplifies to y = 1. This is the solution to the original equation, meaning that when y is equal to 1, the equation -3(y+7) = 2y-26 holds true.
Thus, by performing the division, we have successfully isolated y and found its value. This completes the process of solving the equation. The solution, y = 1, is the value that makes the equation true. To verify this solution, we can substitute y = 1 back into the original equation and check if both sides are equal. This is an important step in problem-solving, as it confirms the accuracy of our calculations and ensures that we have indeed found the correct solution. In the next section, we will perform this verification to ensure the correctness of our answer. Let’s proceed to the verification step.
Step 4: Verify the Solution
After solving for y, it’s crucial to verify the solution to ensure accuracy. Verifying the solution involves substituting the calculated value of y back into the original equation and checking if both sides of the equation are equal. This step acts as a safeguard against potential errors made during the solving process and confirms that the obtained value is indeed the correct solution. It's a fundamental practice in algebra and helps build confidence in your problem-solving abilities. The verification process is straightforward and involves replacing the variable with its solved value and performing the necessary arithmetic operations.
Our solution for the equation -3(y+7) = 2y-26 is y = 1. To verify this, we substitute y = 1 back into the original equation: -3(1+7) = 2(1)-26. Now, we simplify both sides of the equation following the order of operations. On the left side, we first perform the operation inside the parenthesis: 1 + 7 = 8. So, the left side becomes -3(8). Multiplying -3 by 8 gives us -24. Therefore, the left side of the equation simplifies to -24.
On the right side of the equation, we first perform the multiplication: 2(1) = 2. So, the right side becomes 2 - 26. Subtracting 26 from 2 gives us -24. Therefore, the right side of the equation also simplifies to -24. Now we have -24 = -24, which is a true statement. This confirms that our solution, y = 1, is correct. The verification step is essential in mathematics as it provides assurance that the answer is accurate and the equation is balanced. By verifying our solution, we can be confident in our answer and the steps we took to arrive at it. This concludes the detailed step-by-step solution for the given equation.
Conclusion
In conclusion, we have successfully solved for y in the equation -3(y+7) = 2y-26 through a series of methodical steps. We began by distributing the -3 across the terms inside the parenthesis, resulting in the equation -3y - 21 = 2y - 26. Next, we combined like terms by moving the y terms to one side and the constants to the other, which simplified the equation to -5y = -5. Then, we isolated y by dividing both sides by -5, yielding the solution y = 1. Finally, we verified our solution by substituting y = 1 back into the original equation, confirming that both sides were equal. This step-by-step approach not only provides the solution but also reinforces the fundamental principles of algebraic problem-solving.
Understanding and mastering these algebraic techniques is crucial for success in mathematics and various related fields. The ability to solve equations is a foundational skill that underpins more advanced mathematical concepts and has practical applications in numerous disciplines. By breaking down the problem into manageable steps, we have demonstrated a clear and effective method for solving linear equations. Each step, from distribution to combining like terms and isolating the variable, plays a vital role in the process. The verification step further ensures the accuracy of the solution, highlighting the importance of checking one's work.
The process of solving for y in the equation -3(y+7) = 2y-26 exemplifies the systematic approach needed for algebraic problem-solving. By following these steps, students and practitioners can confidently tackle similar equations and build a strong foundation in algebra. This detailed guide serves as a valuable resource for anyone seeking to enhance their understanding of algebraic principles and improve their problem-solving skills. The solution, y = 1, is not just an answer but a testament to the power of methodical application and algebraic reasoning.
