Solving 80 = 3y + 2y + 4 + 1: A Step-by-Step Solution
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In the realm of mathematics, particularly algebra, solving equations is a fundamental skill. This article provides a comprehensive, step-by-step solution to the equation 80 = 3y + 2y + 4 + 1, guiding you through the process with clarity and precision. We will not only arrive at the correct answer but also delve into the underlying principles and techniques involved in solving such equations. Understanding these concepts is crucial for building a strong foundation in algebra and tackling more complex mathematical problems. So, let's embark on this journey of mathematical exploration and unravel the solution to this equation together.
Understanding the Equation: 80 = 3y + 2y + 4 + 1
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Before diving into the solution, let's first understand the equation we are dealing with. The equation 80 = 3y + 2y + 4 + 1 is a linear equation in one variable, 'y'. This means that the highest power of the variable 'y' is 1, and we are looking for a value of 'y' that makes the equation true. In simpler terms, we need to find the value of 'y' that, when substituted into the equation, will make both sides of the equation equal. The left side of the equation is a constant, 80, while the right side involves the variable 'y' along with some constants. Our goal is to isolate 'y' on one side of the equation to determine its value. This involves combining like terms, performing inverse operations, and maintaining the balance of the equation throughout the process. By understanding the structure of the equation, we can approach the solution systematically and confidently. Let's now proceed with the steps involved in solving this equation.
Step 1: Simplify the Equation
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The first step in solving the equation 80 = 3y + 2y + 4 + 1 is to simplify both sides of the equation by combining like terms. Combining like terms involves grouping together terms that have the same variable or are constants. On the right side of the equation, we have two terms with the variable 'y': 3y and 2y. We also have two constant terms: 4 and 1. Combining 3y and 2y gives us 5y. Combining 4 and 1 gives us 5. Therefore, the equation can be simplified as follows:
80 = 3y + 2y + 4 + 1
80 = (3y + 2y) + (4 + 1)
80 = 5y + 5
Now, the equation is in a more simplified form, making it easier to proceed with the next steps. Simplifying the equation is a crucial step in solving algebraic equations as it reduces the complexity and allows us to isolate the variable more effectively. By combining like terms, we have transformed the original equation into a simpler equivalent form, which we can now solve for 'y'. This step demonstrates the importance of algebraic manipulation in simplifying expressions and equations. Let's move on to the next step, where we will further isolate 'y' to find its value.
Step 2: Isolate the Variable Term
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After simplifying the equation to 80 = 5y + 5, the next step is to isolate the variable term, which in this case is 5y. To isolate 5y, we need to eliminate the constant term, which is 5, from the right side of the equation. We can do this by performing the inverse operation. Since 5 is added to 5y, the inverse operation is subtraction. We subtract 5 from both sides of the equation to maintain the equality. This ensures that the equation remains balanced. The steps are as follows:
80 = 5y + 5
Subtract 5 from both sides:
80 - 5 = 5y + 5 - 5
75 = 5y
Now, the equation is further simplified, with the variable term 5y isolated on the right side and the constant 75 on the left side. This step highlights the principle of maintaining balance in an equation by performing the same operation on both sides. Isolating the variable term is a crucial step in solving for the variable, as it brings us closer to determining its value. With the variable term isolated, we can now proceed to the final step, where we will solve for 'y' by dividing both sides of the equation by the coefficient of 'y'.
Step 3: Solve for 'y'
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Having isolated the variable term in the previous step, we now have the equation 75 = 5y. To solve for 'y', we need to get 'y' by itself on one side of the equation. This means we need to eliminate the coefficient 5, which is multiplying 'y'. To do this, we perform the inverse operation, which is division. We divide both sides of the equation by 5 to maintain the equality. The steps are as follows:
75 = 5y
Divide both sides by 5:
75 / 5 = 5y / 5
15 = y
Therefore, the solution to the equation is y = 15. This means that when we substitute 15 for 'y' in the original equation, both sides of the equation will be equal. This step demonstrates the importance of using inverse operations to isolate the variable and solve for its value. We have successfully solved the equation by systematically simplifying it and performing the necessary operations. Let's verify our solution to ensure its accuracy.
Step 4: Verify the Solution
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To ensure that our solution y = 15 is correct, we need to verify it by substituting this value back into the original equation, 80 = 3y + 2y + 4 + 1. If the left side of the equation equals the right side after the substitution, then our solution is correct. Let's perform the substitution:
Original equation: 80 = 3y + 2y + 4 + 1
Substitute y = 15:
80 = 3(15) + 2(15) + 4 + 1
Now, simplify the right side of the equation:
80 = 45 + 30 + 4 + 1
80 = 80
Since the left side of the equation (80) equals the right side of the equation (80) after substituting y = 15, our solution is verified. This step is crucial in solving equations as it confirms the accuracy of our solution and ensures that we have not made any errors in the process. Verification provides confidence in our answer and reinforces the understanding of the equation-solving process. We have successfully solved the equation and verified our solution, demonstrating a thorough understanding of the algebraic principles involved.
Conclusion: The Value of y
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In conclusion, by following a systematic approach involving simplification, isolation of the variable term, and solving for 'y', we have successfully determined the solution to the equation 80 = 3y + 2y + 4 + 1. Our step-by-step guide has shown that the value of y that satisfies this equation is 15. We have also emphasized the importance of verifying the solution to ensure its accuracy. This process not only provides the correct answer but also reinforces the fundamental principles of algebra. Understanding how to solve linear equations is a crucial skill in mathematics and has wide-ranging applications in various fields. By mastering these techniques, you can confidently tackle more complex algebraic problems and build a strong foundation in mathematics. Remember, practice is key to success in mathematics, so continue to solve equations and apply these principles to enhance your understanding and skills.
Selecting the Best Answer
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Having solved the equation 80 = 3y + 2y + 4 + 1 and determined that y = 15, we can now confidently select the best answer from the given options. The options provided were:
A. y = 75
B. y = 1/5
C. y = 15
D. y = -15
Comparing our solution, y = 15, with the given options, it is clear that option C, y = 15, is the correct answer. The other options are incorrect as they do not satisfy the original equation. This final step reinforces the importance of accurately interpreting the solution in the context of the given options. Selecting the correct answer demonstrates a comprehensive understanding of the equation-solving process and the ability to apply the solution to the specific question asked. With the correct answer identified, we have successfully completed the task of solving the equation and selecting the best answer.
Further Practice
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To solidify your understanding of solving linear equations, it is recommended to practice with additional examples. You can find numerous resources online and in textbooks that offer a variety of equations to solve. Remember to follow the steps outlined in this article: simplify the equation, isolate the variable term, solve for the variable, and verify your solution. The more you practice, the more confident and proficient you will become in solving algebraic equations. Consider exploring different types of linear equations, such as those with fractions or decimals, to broaden your skills. Additionally, understanding the applications of linear equations in real-world scenarios can further enhance your appreciation for this fundamental mathematical concept. So, continue to practice and explore the world of algebra!
