Solving And Verifying The Equation (q+2)/7 + (q-2)/2 = 4/7
Hey guys! Today, we're diving into solving a fun little equation and making sure our answer is spot on. We've got this equation: (q+2)/7 + (q-2)/2 = 4/7. Sounds like a puzzle, right? Well, let's break it down step by step. Solving equations might seem intimidating at first, but with a bit of practice and the right approach, you'll be tackling these like a pro. It's all about understanding the underlying principles and applying them systematically. Think of it as building a house – each step relies on the previous one, and a solid foundation is key to a successful outcome. This equation involves fractions, which can sometimes make things look a bit more complex. But don't worry, we'll handle those fractions with ease. Our goal is to isolate the variable 'q' on one side of the equation, so we know exactly what value of 'q' makes the equation true. Once we find that value, we'll plug it back into the original equation to make sure everything checks out. This verification step is super important because it confirms that our solution is correct and that we haven't made any mistakes along the way. It's like double-checking your work on a test – you want to be sure you've got the right answer before moving on. So, let's grab our mathematical toolkits and get started. We'll walk through each step methodically, explaining why we're doing what we're doing. By the end of this article, you'll not only know how to solve this specific equation, but you'll also have a better understanding of how to approach similar problems in the future. Remember, math is a journey, not a destination. It's about learning, growing, and building your skills one step at a time. So, let's embark on this journey together and conquer this equation!
Step-by-Step Solution
Alright, let's get started! The first thing we need to do when we see fractions in an equation is to get rid of them. Fractions can sometimes make the equation look a bit messy, but there's a simple trick to clear them out. We need to find the least common denominator (LCD) of all the fractions in the equation. In our case, we have denominators of 7 and 2. What's the smallest number that both 7 and 2 divide into evenly? That's right, it's 14! The least common denominator (LCD) is a crucial concept when dealing with fractions. It's the smallest multiple that all the denominators share, and it allows us to perform operations like addition and subtraction with ease. Think of it as finding a common language for the fractions so that we can work with them seamlessly. In our equation, the LCD is 14, which means we'll be multiplying both sides of the equation by 14 to eliminate the fractions. This might seem like a big step, but it's actually a very efficient way to simplify the equation and make it easier to solve. By multiplying by the LCD, we ensure that each fraction's denominator will divide evenly into the multiplier, leaving us with whole numbers instead of fractions. This not only simplifies the arithmetic but also reduces the chances of making errors along the way. So, let's proceed with multiplying both sides of the equation by 14. This will set us on the path to isolating 'q' and finding the solution. Remember, the key to success in solving equations is to stay organized and follow the steps carefully. By understanding the underlying principles and applying them systematically, we can tackle even the most complex problems with confidence. So, let's keep moving forward and see how multiplying by the LCD helps us solve this equation!
Now, we're going to multiply both sides of the equation by 14. This means we'll multiply the entire left side, which is (q+2)/7 + (q-2)/2, and the entire right side, which is 4/7, by 14. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. This principle of equality is fundamental in solving equations. It ensures that the equation remains true throughout the process. When we multiply both sides by the same number, we're essentially scaling up the equation without changing its underlying solution. In our case, multiplying by 14 will help us eliminate the fractions and simplify the equation, making it easier to solve for 'q'. So, let's carefully distribute the 14 to each term on both sides of the equation. This involves multiplying 14 by (q+2)/7, 14 by (q-2)/2, and 14 by 4/7. As we perform these multiplications, we'll see how the denominators start to cancel out, leaving us with whole numbers. This is exactly what we want, as it will make the next steps much easier. Remember to pay close attention to the order of operations and the signs of the terms. Accuracy is key in solving equations, and a small mistake can sometimes lead to an incorrect solution. So, let's take our time, double-check our work, and make sure we're doing everything correctly. With a bit of care and attention, we'll be well on our way to solving this equation!
So, after multiplying by 14, we get:
14 * [(q+2)/7 + (q-2)/2] = 14 * (4/7)
Distribute the 14 on the left side:
14 * (q+2)/7 + 14 * (q-2)/2 = 14 * (4/7)
Now, simplify each term:
2(q+2) + 7(q-2) = 2 * 4
See how the fractions are gone? Awesome! Now, let's distribute the numbers outside the parentheses:
2q + 4 + 7q - 14 = 8
Next, we combine like terms on the left side. Like terms are those that have the same variable raised to the same power. In this case, 2q and 7q are like terms, and 4 and -14 are like terms. Combining like terms is a fundamental step in simplifying algebraic expressions and equations. It involves adding or subtracting terms that have the same variable raised to the same power. This process helps to consolidate the expression and make it easier to work with. Think of it as grouping similar items together – you wouldn't mix apples and oranges when counting fruit, right? Similarly, in algebra, we group terms with the same variable and exponent. For example, 3x and 5x are like terms because they both have the variable 'x' raised to the power of 1. However, 3x and 5x^2 are not like terms because they have different exponents. Combining like terms involves adding or subtracting their coefficients, which are the numbers in front of the variables. For instance, when we combine 3x and 5x, we add their coefficients (3 and 5) to get 8x. This simplified term represents the combined value of the original two terms. In our equation, we have the like terms 2q and 7q, which we can combine to get 9q. We also have the like terms 4 and -14, which we can combine to get -10. By combining these like terms, we reduce the number of terms in the equation and make it easier to isolate the variable 'q'. This is a crucial step in solving the equation, as it brings us closer to finding the value of 'q' that satisfies the equation. So, let's proceed with combining the like terms and simplifying the equation further!
(2q + 7q) + (4 - 14) = 8
9q - 10 = 8
Now, we want to isolate the term with 'q' on one side of the equation. To do this, we'll add 10 to both sides:
9q - 10 + 10 = 8 + 10
9q = 18
Finally, to solve for 'q', we divide both sides by 9:
9q / 9 = 18 / 9
q = 2
So, our solution is q = 2. But hold on, we're not done yet! We need to check our answer to make sure it's correct.
Checking the Answer
Okay, we've got our potential solution, q = 2. Now comes the super important part: checking our answer. Why is this so crucial? Well, it's like proofreading an essay or testing a recipe. We want to make sure we haven't made any mistakes along the way and that our solution actually works in the original equation. Checking our answer involves plugging the value we found for 'q' (which is 2 in this case) back into the original equation. We then simplify both sides of the equation and see if they are equal. If they are, then our solution is correct! If they're not, then we know we've made a mistake somewhere and need to go back and review our steps. This process of verification is a cornerstone of mathematical problem-solving. It's not just about getting an answer; it's about ensuring the answer is accurate and valid. It helps us build confidence in our work and develop a deeper understanding of the mathematical concepts involved. So, let's carefully substitute q = 2 into the original equation and see what happens. We'll follow the order of operations, simplifying each side step by step. If the left side equals the right side, then we can confidently say that q = 2 is the correct solution. And if not, we'll know it's time to put on our detective hats and find the error. Remember, the goal isn't just to get the answer; it's to get the correct answer and understand why it's correct. So, let's dive into the verification process and give our solution a thorough check!
Let's plug q = 2 back into the original equation:
(q+2)/7 + (q-2)/2 = 4/7
(2+2)/7 + (2-2)/2 = 4/7
Simplify:
4/7 + 0/2 = 4/7
4/7 + 0 = 4/7
4/7 = 4/7
It checks out! Both sides of the equation are equal. That means our solution, q = 2, is correct.
Final Answer
Woohoo! We did it! We successfully solved the equation and verified our answer. The solution set is {2}.
So, the correct choice is:
A. The solution set is { 2 }.
(Type an integer or a simplified fraction.)
Great job, guys! You've taken on a tricky equation and come out on top. Remember, practice makes perfect, so keep solving those equations, and you'll become a math whiz in no time!
