Solve For S: A Step-by-Step Guide

Hey guys! Ever found yourself staring at an equation and wondering, "How do I even begin to solve this?" Well, you're not alone! Today, we're going to break down a classic algebraic equation and show you exactly how to solve for the variable s. We'll take it step-by-step, so even if algebra feels like a foreign language right now, you'll be fluent by the end of this guide. So, let's dive into this equation: (5/8) - (3/8)s = 2 - s

Understanding the Equation

Before we start crunching numbers, let's understand what this equation is telling us. We have a variable, s, which represents an unknown value. Our goal is to isolate s on one side of the equation so we can figure out what it equals. The equation involves fractions, subtraction, and a combination of constants (like 5/8 and 2) and terms with the variable s (like -3/8 s and -s). Don't let the fractions scare you! We'll tackle them together. Remember, the key to solving any equation is to perform the same operations on both sides to maintain balance. Think of it like a scale: whatever you add or subtract from one side, you have to do the same on the other to keep it level. This principle is the foundation of algebraic manipulation.

The first step in solving this equation involves dealing with those fractions. Fractions can sometimes make things look more complicated than they really are, so let's eliminate them. To do this, we'll find the least common multiple (LCM) of the denominators in our equation. In this case, we only have one denominator: 8. So, the LCM is simply 8. Now, we'll multiply every term in the equation by 8. This is crucial – we need to multiply every term, both on the left-hand side and the right-hand side, to maintain the balance of the equation. This step might seem a bit tedious, but it's a powerful technique for simplifying equations with fractions. By multiplying each term by the LCM, we effectively clear the fractions, making the equation much easier to work with. Let's see how this looks in practice. We'll multiply (5/8) by 8, (-3/8)s by 8, 2 by 8, and -s by 8. This will give us a new equation without any fractions, which will be much simpler to solve. This is a common and very effective strategy in algebra, so it's worth mastering this technique.

Step 1: Clearing the Fractions

Okay, let's get rid of those fractions! As we discussed, we'll multiply both sides of the equation by 8. This means we're doing the following:

8 * [(5/8) - (3/8)s] = 8 * [2 - s]

Now, we need to distribute the 8 on both sides. Remember the distributive property? It means we multiply the 8 by each term inside the parentheses. So, on the left side, we have:

8 * (5/8) - 8 * (3/8)s

And on the right side:

8 * 2 - 8 * s

Let's simplify each term. 8 * (5/8) becomes 5 (the 8s cancel out!). 8 * (3/8)s becomes 3s (again, the 8s cancel). 8 * 2 is simply 16, and 8 * s is 8s. So, our equation now looks like this:

5 - 3s = 16 - 8s

See how much cleaner that looks? By multiplying through by the least common multiple, we've transformed the equation from one filled with fractions to a much simpler linear equation. This is a huge step forward in solving for s. Now, we can move on to the next phase: isolating the terms with s on one side of the equation.

Step 2: Isolating the 's' Terms

The next crucial step is to gather all the terms containing our variable, s, on one side of the equation. It doesn't matter which side we choose – we can collect them on the left or the right. However, a good strategy is to move the terms in a way that results in a positive coefficient for s. This can make the next steps a little easier. Looking at our current equation, 5 - 3s = 16 - 8s, we see that we have -3s on the left and -8s on the right. To get a positive coefficient, we can add 8s to both sides. This will eliminate the -8s term on the right and give us a positive s term on the left.

So, let's add 8s to both sides:

5 - 3s + 8s = 16 - 8s + 8s

Now, simplify by combining like terms. On the left, -3s + 8s combines to 5s. On the right, -8s + 8s cancels out, leaving us with just 16. Our equation now looks like this:

5 + 5s = 16

We've successfully gathered all the s terms on the left side. Now, we need to isolate the s term completely by getting rid of the constant term (the 5) on the left side. This is the next logical step in our journey to solve for s. Remember, we're getting closer and closer to our goal with each step!

Step 3: Isolating 's' Completely

Alright, we're making great progress! We've managed to get all the s terms on one side of the equation. Our equation currently looks like this: 5 + 5s = 16. Now, we need to isolate the s term completely. This means we need to get rid of that pesky 5 that's being added to 5s on the left side. To do this, we'll use the inverse operation. Since 5 is being added, we'll subtract 5 from both sides of the equation. Remember, maintaining balance is key! Whatever we do to one side, we must do to the other.

So, let's subtract 5 from both sides:

5 + 5s - 5 = 16 - 5

Now, simplify. On the left side, 5 - 5 cancels out, leaving us with just 5s. On the right side, 16 - 5 is 11. Our equation now looks like this:

5s = 11

We're almost there! The s term is now almost completely isolated. We just have one more step to take: getting rid of the coefficient (the number multiplying s). In this case, 5 is multiplying s. So, how do we undo multiplication? You guessed it: we divide!

Step 4: Solving for 's'

We've reached the final step! Our equation is now 5s = 11. The only thing standing between us and the value of s is that 5 that's multiplying it. As we discussed, to undo multiplication, we divide. So, we'll divide both sides of the equation by 5. This will isolate s and give us our solution.

Let's divide both sides by 5:

(5s) / 5 = 11 / 5

Now, simplify. On the left side, the 5s cancel out, leaving us with just s. On the right side, 11 / 5 is a fraction that we can leave as is (or convert to a decimal if we prefer). So, our solution is:

s = 11/5

Or, if you prefer a decimal representation:

s = 2.2

And there you have it! We've successfully solved for s. We took it step-by-step, clearing fractions, isolating terms, and using inverse operations. Remember, the key to solving algebraic equations is to stay organized, maintain balance, and tackle each step methodically. You can apply these techniques to a wide range of equations. Practice makes perfect, so don't be afraid to try more examples and build your skills. You've got this!

Final Answer

So, the final answer is:

s = 11/5 or 2.2