Solve For C: A Step-by-Step Guide

Hey everyone! Today, we are diving into a fundamental algebraic problem: solving for the variable $c$ in the equation $-\frac{c}{2} + 43 = 50$. This is a common type of problem you'll encounter in algebra, and mastering it is essential for tackling more complex equations later on. So, let's break it down step-by-step and make sure we understand every part of the process. Whether you're a student just starting algebra or someone looking to refresh your skills, this guide will walk you through the solution clearly and effectively. Let's get started and unravel this equation together!

Understanding the Equation

Alright, let's start by understanding the equation we're working with: $-\frac{c}{2} + 43 = 50$. In this equation, $c$ is our variable, which means it's the unknown value we're trying to find. The equation is essentially saying that if we take $c$, divide it by -2 (which is the same as multiplying by $-\frac{1}{2}$), and then add 43, the result will be 50. Our goal is to isolate $c$ on one side of the equation so we can determine its value. Think of it like peeling back layers to get to the core. Each step we take will bring us closer to having $c$ all by itself on one side, giving us the solution we need. We'll use algebraic principles to manipulate the equation while keeping it balanced. This means whatever we do on one side, we have to do on the other side as well. This keeps the equation true and allows us to solve for $c$ accurately. So, let's jump into the first step and start solving this equation together!

Step 1: Isolate the Term with $c$

The first step in solving for $c$ is to isolate the term that contains $c$, which in this case is $-\frac{c}{2}$. To do this, we need to get rid of the +43 that's on the same side of the equation. The way we do this is by performing the opposite operation. Since 43 is being added, we'll subtract 43 from both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced. So, we have:

$-\frac{c}{2} + 43 - 43 = 50 - 43$

Simplifying this, we get:

$-\frac{c}{2} = 7$

Now, we've successfully isolated the term with $c$ on one side of the equation. This is a significant step forward because we're getting closer to having $c$ by itself. By subtracting 43 from both sides, we've maintained the equation's balance and moved one step closer to our goal. Next, we'll need to deal with the fraction that's attached to $c$. This will involve another operation that will help us completely isolate $c$ and find its value. Keep going, guys, we're making great progress!

Step 2: Solve for $c$

Now that we have $-\frac{c}{2} = 7$, our next step is to completely isolate $c$. Currently, $c$ is being divided by -2. To undo this division, we need to perform the inverse operation, which is multiplication. We'll multiply both sides of the equation by -2. Again, it's crucial to do the same thing on both sides to maintain the balance of the equation. This gives us:

$(-2) imes (-\frac{c}{2}) = 7 imes (-2)$

On the left side, multiplying $-\frac{c}{2}$ by -2 cancels out the -2 in the denominator, leaving us with just $c$. On the right side, 7 multiplied by -2 equals -14. So, the equation simplifies to:

$c = -14$

And there you have it! We've successfully solved for $c$. The value of $c$ that makes the original equation true is -14. This step involved using the principle of inverse operations to undo the division and isolate our variable. By multiplying both sides by -2, we effectively "canceled out" the denominator and revealed the value of $c$. We're now confident that we have the correct solution. But, just to be absolutely sure, let's do a quick check to verify our answer.

Step 3: Check the Solution

To ensure our solution is correct, we need to substitute $c = -14$ back into the original equation and see if it holds true. Our original equation was:

$-\frac{c}{2} + 43 = 50$

Now, let's replace $c$ with -14:

$-\frac{-14}{2} + 43 = 50$

First, we simplify the fraction. Dividing -14 by 2 gives us -7. So, the equation becomes:

$-(-7) + 43 = 50$

A negative of a negative number is positive, so -(-7) becomes 7. Now we have:

$7 + 43 = 50$

Adding 7 and 43, we get:

$50 = 50$

Since the left side of the equation equals the right side, our solution is correct! This check confirms that $c = -14$ is indeed the value that satisfies the original equation. Checking our solution is a crucial step in problem-solving because it helps us catch any mistakes we might have made along the way. It gives us confidence in our answer and ensures that we're on the right track. So, always remember to check your solutions whenever you can. Great job, guys! We've not only solved for $c$ but also verified our solution.

Conclusion

Awesome work, everyone! We've successfully solved for $c$ in the equation $-\frac{c}{2} + 43 = 50$. We found that $c = -14$. We started by isolating the term with $c$, then we solved for $c$ by using inverse operations, and finally, we checked our solution to make sure it was correct. This problem illustrates the fundamental principles of solving algebraic equations, and these steps can be applied to many similar problems. Remember, the key is to keep the equation balanced by performing the same operations on both sides and to use inverse operations to isolate the variable you're solving for. Practice makes perfect, so keep working on these types of problems, and you'll become more confident in your algebra skills. Whether you're tackling homework, preparing for a test, or just expanding your mathematical knowledge, understanding how to solve for variables is a valuable skill. So, keep up the great work, and you'll be solving even more complex equations in no time!