Solving For U A Step By Step Guide

Are you struggling with solving algebraic equations? Don't worry, guys, you're not alone! Many people find algebra challenging, but with the right approach, it can become much easier. In this comprehensive guide, we'll break down the steps to solve the equation -(5/3)u + 3 = -(6/5)u - (6/5) for the variable 'u'. We'll provide a clear, step-by-step explanation, making it easy for you to follow along and master this type of problem. So, let's dive in and conquer this equation together!

Understanding the Equation: A Crucial First Step

Before we start crunching numbers, it's essential to understand what the equation -(5/3)u + 3 = -(6/5)u - (6/5) actually means. At its core, an equation is a statement that two expressions are equal. In this case, the expression on the left side of the equals sign (-(5/3)u + 3) is equal to the expression on the right side (-(6/5)u - (6/5)). Our goal is to find the value of 'u' that makes this statement true. The variable 'u' represents an unknown quantity, and by solving the equation, we're essentially uncovering its value. To solve for 'u', we need to isolate it on one side of the equation. This involves performing algebraic operations, such as addition, subtraction, multiplication, and division, on both sides of the equation to maintain the balance. Remember, whatever you do to one side, you must do to the other to keep the equation valid. With a solid grasp of this fundamental concept, we can confidently move on to the next step: simplifying the equation.

Step 1: Simplifying the Equation for Clarity

The equation -(5/3)u + 3 = -(6/5)u - (6/5), while solvable in its current form, can be a bit intimidating to look at. Fractions often add a layer of complexity, making it harder to visualize the steps involved. Therefore, our first step is to simplify the equation by eliminating these fractions. To do this, we'll find the least common multiple (LCM) of the denominators in the equation. In this case, the denominators are 3 and 5. The LCM of 3 and 5 is 15. Now, we'll multiply both sides of the equation by 15. This is a crucial step because multiplying by the LCM will clear out the fractions, making the equation much easier to work with. Let's break it down. Multiplying the left side, 15 * (-(5/3)u + 3), we distribute the 15 to both terms: 15 * (-(5/3)u) + 15 * 3. This simplifies to -25u + 45. Similarly, multiplying the right side, 15 * (-(6/5)u - (6/5)), we distribute the 15: 15 * (-(6/5)u) - 15 * (6/5). This simplifies to -18u - 18. So, after multiplying both sides by 15, our equation transforms into the much cleaner form: -25u + 45 = -18u - 18. This simplified equation is equivalent to the original one, but it's free of fractions, making it much easier to manipulate and solve for 'u'.

Step 2: Isolating the 'u' Terms: Bringing Like Terms Together

Now that we've successfully simplified the equation to -25u + 45 = -18u - 18, the next step is to isolate the terms containing 'u' on one side of the equation. This is a fundamental strategy in solving algebraic equations, as it brings us closer to our goal of finding the value of 'u'. To do this, we need to move all the 'u' terms to one side and all the constant terms (the numbers without 'u') to the other side. Let's start by moving the '-18u' term from the right side to the left side. To do this, we'll add 18u to both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain the balance. Adding 18u to both sides gives us: -25u + 18u + 45 = -18u + 18u - 18. This simplifies to -7u + 45 = -18. Now, we need to move the constant term, '+45', from the left side to the right side. To do this, we'll subtract 45 from both sides of the equation: -7u + 45 - 45 = -18 - 45. This simplifies to -7u = -63. We've now successfully isolated the 'u' term on the left side, making the equation much closer to being solved.

Step 3: Solving for 'u': The Final Division

We've made excellent progress! Our equation is now in the simplified form -7u = -63. This means that -7 multiplied by 'u' is equal to -63. To finally solve for 'u', we need to undo this multiplication. The inverse operation of multiplication is division, so we'll divide both sides of the equation by -7. This will isolate 'u' on the left side and give us its value. Dividing both sides by -7 gives us: (-7u) / -7 = (-63) / -7. On the left side, the -7 in the numerator and the -7 in the denominator cancel each other out, leaving us with just 'u'. On the right side, -63 divided by -7 equals 9. Remember, a negative number divided by a negative number results in a positive number. Therefore, the equation simplifies to u = 9. Congratulations! We've successfully solved for 'u'. This means that the value of 'u' that makes the original equation true is 9. To be absolutely sure of our answer, it's always a good idea to check our solution.

Step 4: Verifying the Solution: Ensuring Accuracy

We've arrived at the solution u = 9, but before we declare victory, it's crucial to verify our answer. This step is essential to ensure that our solution is accurate and that we haven't made any errors in our calculations. To verify, we'll substitute 'u = 9' back into the original equation, -(5/3)u + 3 = -(6/5)u - (6/5), and see if both sides of the equation are equal. Substituting 'u = 9' into the left side of the equation, -(5/3)u + 3, we get: -(5/3) * 9 + 3. This simplifies to -15 + 3, which equals -12. Now, let's substitute 'u = 9' into the right side of the equation, -(6/5)u - (6/5), we get: -(6/5) * 9 - (6/5). This simplifies to -54/5 - 6/5, which equals -60/5, which further simplifies to -12. Comparing both sides, we see that the left side (-12) is equal to the right side (-12). This confirms that our solution, u = 9, is indeed correct. Verifying our solution provides us with confidence in our answer and reinforces our understanding of the problem-solving process. It's a valuable habit to develop in mathematics and in any problem-solving situation.

Conclusion: Mastering Algebraic Equations

We've successfully navigated the steps to solve the equation -(5/3)u + 3 = -(6/5)u - (6/5) and found that u = 9. We started by understanding the equation, then simplified it by eliminating fractions. We isolated the 'u' terms, solved for 'u' through division, and finally, verified our solution to ensure accuracy. This step-by-step approach can be applied to a wide range of algebraic equations. Remember, guys, practice is key to mastering algebra. The more you solve equations, the more comfortable and confident you'll become. Don't be afraid to tackle challenging problems, and always remember to break them down into smaller, manageable steps. With persistence and a clear understanding of the principles involved, you can conquer any algebraic equation that comes your way! So keep practicing, keep learning, and keep solving!