Solving Equations: Find 'c' In 4(3+c)=3(c-3)

Hey guys! Let's dive into a fun little math problem today. We're going to tackle the equation 4(3+c)=3(c-3) and figure out what 'c' is. Math can seem intimidating sometimes, but trust me, it's like a puzzle – and we're about to solve it together! So, grab your thinking caps, and let’s get started on this mathematical adventure!

Breaking Down the Equation

Okay, first things first, let's take a good look at our equation: 4(3+c)=3(c-3). The goal here is to isolate 'c' on one side of the equation. To do that, we'll need to use some algebraic techniques. Don’t worry; it’s not as scary as it sounds! We'll go through it step by step, making sure everything is clear as we go. Remember, math is all about taking things one step at a time, and you've totally got this!

Step 1: Distribute

The first thing we need to do is get rid of those parentheses. How do we do that? By using the distributive property. This means we multiply the number outside the parentheses by each term inside the parentheses. So, on the left side of the equation, we have 4 multiplied by both 3 and 'c'. That gives us 4 * 3 = 12 and 4 * c = 4c. So, the left side becomes 12 + 4c.

Now, let's do the same on the right side. We have 3 multiplied by both 'c' and -3. That gives us 3 * c = 3c and 3 * -3 = -9. So, the right side becomes 3c - 9. Now our equation looks like this:

12 + 4c = 3c - 9

See? We've already made progress! By distributing, we've simplified the equation and made it easier to work with. This is a crucial step in solving for 'c', so make sure you're comfortable with the distributive property. It's a fundamental tool in algebra, and you'll be using it a lot!

Step 2: Gather Like Terms

The next step is to gather all the 'c' terms on one side of the equation and all the constant terms (the numbers without 'c') on the other side. This helps us isolate 'c' and get closer to our solution. To do this, we'll use the properties of equality, which basically say that we can add or subtract the same thing from both sides of the equation without changing the equation's balance. It’s like keeping a scale balanced – whatever you do to one side, you have to do to the other.

Let’s start by moving the 'c' terms. We have 4c on the left and 3c on the right. To get the 3c to the left side, we can subtract 3c from both sides. This gives us:

12 + 4c - 3c = 3c - 9 - 3c

Simplifying this, we get:

12 + c = -9

Now, let's move the constant terms. We have 12 on the left and -9 on the right. To get the 12 to the right side, we subtract 12 from both sides:

12 + c - 12 = -9 - 12

Simplifying this, we get:

c = -21

Ta-da! We've gathered the like terms and isolated 'c'. This step is all about rearranging the equation to make it easier to solve. By adding and subtracting the same terms from both sides, we maintain the equation's balance while moving the terms where we need them. Now, we're just one step away from our final answer!

Step 3: Solve for 'c'

We're almost there! We've simplified the equation to c = -21. Guess what? We've already solved for 'c'! That’s right, after all that rearranging and simplifying, we found that 'c' is equal to -21. This is the moment of truth, the answer we've been working towards. Solving for a variable often involves a series of steps, but sometimes, like in this case, the final simplification reveals the answer directly.

So, to recap, we started with a slightly complex equation, used the distributive property to get rid of the parentheses, gathered like terms by adding and subtracting from both sides, and finally, we found that c = -21. See? Not so scary after all!

Checking Our Solution

Now, before we declare victory, it's always a good idea to check our answer. This is like double-checking your work on a puzzle to make sure all the pieces fit. To check our solution, we'll plug c = -21 back into the original equation: 4(3+c)=3(c-3).

Plugging it In

Let's substitute -21 for 'c' in the original equation:

4(3 + (-21)) = 3((-21) - 3)

Now, let's simplify each side.

Simplifying the Left Side

First, we simplify inside the parentheses: 3 + (-21) = -18. Then, we multiply by 4: 4 * (-18) = -72. So, the left side of the equation simplifies to -72.

Simplifying the Right Side

Similarly, we simplify inside the parentheses on the right side: (-21) - 3 = -24. Then, we multiply by 3: 3 * (-24) = -72. So, the right side of the equation simplifies to -72.

The Moment of Truth

We found that both sides of the equation simplify to -72. This means that our solution, c = -21, is correct! Checking our solution is a great way to ensure accuracy and build confidence in our mathematical skills. It's like the final piece of the puzzle clicking into place.

Wrapping Up

And there you have it, folks! We successfully solved the equation 4(3+c)=3(c-3) and found that c = -21. We broke down the problem into manageable steps, used the distributive property, gathered like terms, and checked our answer. Math is all about problem-solving, and you guys just rocked it!

Remember, the key to mastering math is practice and persistence. Don't be afraid to tackle challenging problems, and always double-check your work. With a little effort and the right approach, you can conquer any equation that comes your way. Keep practicing, and you'll become math whizzes in no time! Keep shining, mathletes! You've got this! ✨