Solving For 'u': A Step-by-Step Guide

Hey guys! Let's dive into solving equations, specifically when we're trying to figure out the value of a variable. Today, our mission is to tackle this equation: $\frac{5}{3u} - \frac{2}{7u} = 1$. Don't worry, it might look a bit intimidating at first, but trust me, it's totally manageable. We'll break it down step by step, making sure we understand every move. Solving for a variable, like 'u' in this case, is a fundamental skill in algebra, and it's super important for more advanced math stuff later on. So, let's get started and make sure we nail this concept. Ready? Let's do it!

Understanding the Equation

Alright, before we jump into the math, let's quickly understand what we're dealing with. The equation $\frac{5}{3u} - \frac{2}{7u} = 1$ is an algebraic equation. Our primary goal is to isolate 'u' on one side of the equation to find its value. This equation involves fractions with 'u' in the denominator, which adds a little twist, but we'll handle it like champs. It's all about applying the right operations – addition, subtraction, multiplication, and division – in the correct order to simplify the equation and eventually solve for 'u'. Think of it like a puzzle; each step brings us closer to the final solution. Remembering the basics of fractions is a good idea too! We're going to be using common denominators, and we'll be dealing with numerators and denominators, which are crucial when performing operations with fractions. So, basically, we need to figure out what value 'u' has to be for the whole equation to be true. This might seem challenging now, but I promise that as we go through the steps together, it will become crystal clear.

Identifying the Terms and Variables

Okay, let's break this equation down further. The equation $\frac{5}{3u} - \frac{2}{7u} = 1$ has fractions. In the fractions, we have variables in the denominator. A variable is a symbol, in this case 'u', that represents an unknown value. The terms in our equation are $\frac{5}{3u}$, $-\frac{2}{7u}$, and $1$. The term '1' stands alone, but the other two involve 'u'. Our goal is to find the exact value of 'u' that makes this equation true. We will make sure we pay close attention to how to get rid of the fractions and move 'u' from the denominators to a more accessible place. This is an important first step because identifying the pieces helps us to know how to proceed with solving the equation. Understanding the role of each term and the purpose of the variable makes the solution process easier to understand and execute. Now, let's start with the solution process!

Step-by-Step Solution

Now, for the main event: solving the equation! This is where we get our hands dirty with some calculations. I'll walk you through each step, ensuring you understand the logic behind every move. We'll be making sure we perform the operations correctly and are careful to avoid making any mistakes. It's like a recipe; follow the steps carefully, and you'll get the desired result. Let's get to it.

Finding a Common Denominator

First things first, we need to deal with those fractions. Remember how to add or subtract fractions? Yep, we need a common denominator! Looking at the denominators $3u$ and $7u$, we want the smallest number that both $3$ and $7$ divide into evenly. The least common multiple (LCM) of 3 and 7 is 21. So, our common denominator will be $21u$. We're essentially finding a number that both denominators can easily be converted into. We must make sure each fraction has the same denominator before we combine them. Doing this helps us to simplify the equation and reduce the number of terms. This is a critical step because it allows us to combine the fractions into a single term, making our equation easier to solve. It's a key component in simplifying and ultimately solving for u.

Rewriting the Fractions

Now, let's rewrite our fractions so that they have the common denominator of $21u$. We'll multiply the numerator and denominator of each fraction by a number that gets the denominator to $21u$.

• For the first fraction, $\frac{5}{3u}$, we multiply both the numerator and denominator by 7: $\frac{5 \cdot 7}{3u \cdot 7} = \frac{35}{21u}$.
• For the second fraction, $\frac{2}{7u}$, we multiply both the numerator and denominator by 3: $\frac{2 \cdot 3}{7u \cdot 3} = \frac{6}{21u}$.

Now, our equation becomes $\frac{35}{21u} - \frac{6}{21u} = 1$. See, now we are getting somewhere. We changed the form of the fractions without changing their value. We're simply expressing them in a way that makes it easy to perform operations.

Combining the Fractions

Since our fractions now have the same denominator, we can combine them! Subtract the numerators and keep the common denominator. So, $\frac{35}{21u} - \frac{6}{21u}$ becomes $\frac{35 - 6}{21u}$, which simplifies to $\frac{29}{21u}$. Our equation is now $\frac{29}{21u} = 1$. See how much simpler it looks now? Combining like terms simplifies equations and makes it easier to isolate the variable. This is where the equation becomes more manageable, and our progress becomes more obvious.

Isolating 'u'

Our next objective is to get 'u' alone on one side of the equation. To do this, we will multiply both sides of the equation by $21u$. This gives us $29 = 21u$. Now, we're getting really close! To isolate 'u', we now divide both sides of the equation by 21. So, $\frac{29}{21} = \frac{21u}{21}$, which simplifies to $\frac{29}{21} = u$. So, we found that $u = \frac{29}{21}$. We did it, guys!

Checking Your Answer

Alright, let's make sure we didn't make any mistakes. It's always a good idea to check your answer to verify it. We can substitute the value of 'u' we found back into the original equation: $\frac{5}{3u} - \frac{2}{7u} = 1$.

Substitute $\frac{29}{21}$ for u: $\frac{5}{3 \cdot \frac{29}{21}} - \frac{2}{7 \cdot \frac{29}{21}} = 1$.

Simplify each fraction by multiplying:

• $\frac{5}{\frac{87}{21}} - \frac{2}{\frac{203}{21}} = 1$
• $\frac{5 \cdot 21}{87} - \frac{2 \cdot 21}{203} = 1$
• $\frac{105}{87} - \frac{42}{203} = 1$
• $\frac{35}{29} - \frac{6}{29} = 1$
• $\frac{29}{29} = 1$
• $1 = 1$

Since the equation is true, we know we are correct! Hooray!

Conclusion

And there you have it! We successfully solved for 'u' in the equation $\frac{5}{3u} - \frac{2}{7u} = 1$. We started with a seemingly complex equation and broke it down into smaller, manageable steps. We identified the terms and variables, found a common denominator, rewrote the fractions, combined like terms, isolated 'u', and then, we checked our answer. Remember, practice is key! The more you solve equations like this, the more comfortable you'll become. You'll start to recognize the patterns and steps, and the process will become much easier. Keep up the great work, and don't be afraid to try new problems. You've got this!