Solving For Y Step By Step Guide (7/2)y - 3 = (-4/5)y - (5/2)
Hey guys! Are you struggling with solving equations? No worries, we've all been there. In this article, we're going to break down how to solve the equation (7/2)y - 3 = (-4/5)y - (5/2) step-by-step. Solving for a variable, especially when fractions are involved, might seem daunting at first, but with a clear understanding of the process, it becomes quite manageable. This comprehensive guide aims to provide you with the tools and knowledge to tackle such problems confidently. We will cover each step in detail, ensuring that you not only get the correct answer but also understand the underlying principles of algebraic manipulation. Let's dive in and make math a little less intimidating and a lot more fun!
Understanding the Basics of Algebraic Equations
Before we jump into the specific equation, let's quickly recap the basics of algebraic equations. An equation is a mathematical statement that asserts the equality of two expressions. Our goal in solving an equation is to isolate the variable (in this case, y) on one side of the equation. This involves performing operations on both sides of the equation to maintain the balance. The key principle here is that whatever you do to one side, you must do to the other. This ensures that the equality remains true throughout the solving process. We'll be using several techniques, including combining like terms, adding or subtracting the same value from both sides, and multiplying or dividing both sides by the same non-zero value. These are the fundamental tools in our algebraic toolkit, and mastering them will significantly improve your ability to solve a wide range of equations. Remember, the goal is to simplify the equation step by step until we have y all by itself on one side, giving us its value.
The Importance of Order of Operations
When dealing with algebraic equations, especially those involving fractions and multiple terms, adhering to the order of operations is crucial. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is often used as a guide. However, in solving equations, we often reverse this order to isolate the variable effectively. We typically deal with addition and subtraction first, followed by multiplication and division. This approach helps in peeling away the layers surrounding the variable, gradually simplifying the equation. It's also important to remember that when we perform an operation on one side of the equation, we must perform the same operation on the other side to maintain balance. This principle is the cornerstone of solving equations, ensuring that the equality remains valid throughout the process. Understanding and applying the correct order of operations is key to avoiding common mistakes and arriving at the correct solution. So, let's keep this in mind as we tackle our equation!
Step-by-Step Solution
Okay, let's get down to business and solve this equation together! Here's the equation we're tackling:
(7/2)y - 3 = (-4/5)y - (5/2)
Step 1: Get All y Terms on One Side
Our first goal is to gather all the terms containing y on one side of the equation. To do this, we'll add (4/5)y to both sides. This will eliminate the y term from the right side and bring it over to the left side. Remember, whatever we do to one side, we must do to the other to keep the equation balanced.
(7/2)y - 3 + (4/5)y = (-4/5)y - (5/2) + (4/5)y
Simplifying this, we get:
(7/2)y + (4/5)y - 3 = -5/2
Now, we need to combine the y terms. To add fractions, we need a common denominator. The least common multiple of 2 and 5 is 10. So, we'll convert both fractions to have a denominator of 10:
((7/2) * (5/5))y + ((4/5) * (2/2))y - 3 = -5/2
This gives us:
(35/10)y + (8/10)y - 3 = -5/2
Combining the fractions, we have:
(43/10)y - 3 = -5/2
Great job! We've successfully moved all the y terms to one side. Now, let's move on to the next step.
Step 2: Move Constant Terms to the Other Side
Now that we have all the y terms on the left side, let's move the constant terms (the numbers without y) to the right side. We currently have a -3 on the left side, so we'll add 3 to both sides to eliminate it from the left:
(43/10)y - 3 + 3 = -5/2 + 3
This simplifies to:
(43/10)y = -5/2 + 3
To add -5/2 and 3, we need to express 3 as a fraction with a denominator of 2. So, 3 becomes 6/2:
(43/10)y = -5/2 + 6/2
Adding the fractions, we get:
(43/10)y = 1/2
Excellent! We're getting closer to isolating y. Now, let's move on to the final step.
Step 3: Isolate y
We're almost there! We have (43/10)y = 1/2. To isolate y, we need to get rid of the 43/10 that's multiplying y. We can do this by multiplying both sides of the equation by the reciprocal of 43/10, which is 10/43:
(10/43) * (43/10)y = (1/2) * (10/43)
On the left side, (10/43) * (43/10) cancels out, leaving us with just y:
y = (1/2) * (10/43)
Now, we multiply the fractions on the right side:
y = 10/86
Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
y = (10 ÷ 2) / (86 ÷ 2)
y = 5/43
And that's it! We've solved for y. The solution is y = 5/43. Give yourself a pat on the back!
Alternative Methods for Solving the Equation
While we've walked through one method to solve the equation, it's worth noting that there are often multiple paths to the same destination in mathematics. One alternative approach involves eliminating the fractions early on in the process. This can simplify the equation and make it easier to work with, especially for those who find fractions a bit tricky. To eliminate fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators. In our case, the denominators are 2 and 5, so the LCM is 10. Multiplying both sides of the original equation by 10 would clear the fractions, giving us a cleaner equation to solve. This method is particularly useful when dealing with equations that have several fractional terms, as it streamlines the process and reduces the chances of making errors in fraction arithmetic. Remember, the goal is to choose the method that you find most comfortable and efficient, and understanding different approaches can help you develop a deeper understanding of the underlying mathematical principles.
Clearing Fractions by Multiplying by the LCM
As mentioned earlier, one effective strategy for simplifying equations with fractions is to clear the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. This method can often make the equation easier to handle, as it eliminates the need to work with fractions directly. Let's briefly illustrate how this would work with our equation. The original equation is (7/2)y - 3 = (-4/5)y - (5/2). The denominators are 2 and 5, and their LCM is 10. By multiplying every term in the equation by 10, we can eliminate the fractions. This technique is particularly valuable when dealing with more complex equations involving multiple fractions, as it can significantly reduce the complexity of the calculations. Clearing fractions is a powerful tool in your algebraic arsenal, and mastering it can make solving equations a much smoother process. Remember to always ensure that you multiply every term on both sides of the equation by the LCM to maintain balance and accuracy.
Common Mistakes to Avoid
When solving equations, it's easy to make small mistakes that can lead to incorrect answers. Let's go over some common pitfalls to watch out for. One frequent mistake is forgetting to distribute a number when multiplying it across parentheses or multiple terms. For example, if you have 2(x + 3), you need to multiply both the x and the 3 by 2. Another common error is not applying the same operation to both sides of the equation. Remember, the golden rule is that whatever you do to one side, you must do to the other to maintain balance. Additionally, mistakes often occur when dealing with negative signs. Be extra careful when adding, subtracting, multiplying, or dividing with negative numbers. It's also crucial to double-check your work, especially when dealing with fractions. Make sure you've found common denominators correctly and simplified fractions properly. By being aware of these common mistakes and taking your time to double-check your steps, you can significantly improve your accuracy in solving equations. Practice makes perfect, so keep at it!
The Importance of Checking Your Answer
One of the best habits you can develop when solving equations is to check your answer. Plugging your solution back into the original equation can quickly reveal whether you've made a mistake along the way. This step is particularly important when dealing with complex equations or when you've used multiple steps to arrive at your solution. To check your answer, substitute the value you found for y back into the original equation, (7/2)y - 3 = (-4/5)y - (5/2). If both sides of the equation are equal after the substitution, then your solution is correct. If the two sides are not equal, it indicates that there's an error somewhere in your calculations, and you'll need to go back and review your steps. Checking your answer is not just about verifying your solution; it's also a valuable learning opportunity. By identifying and correcting your mistakes, you reinforce your understanding of the equation-solving process. So, make it a habit to always check your answers – it's a small step that can make a big difference in your mathematical success!
Conclusion
So there you have it! We've successfully solved for y in the equation (7/2)y - 3 = (-4/5)y - (5/2), and found that y = 5/43. We walked through each step, from combining like terms to isolating the variable, and even discussed alternative methods and common mistakes to avoid. Solving equations can be challenging, but with a solid understanding of the basic principles and a bit of practice, you can tackle even the trickiest problems. Remember, the key is to stay organized, double-check your work, and don't be afraid to make mistakes – they're part of the learning process! Keep practicing, and you'll become a pro at solving equations in no time. You've got this!
