Solve For C In 61 = -9c + 7

Hey math whizzes! Ever stared at an equation and thought, "What in the world is 'c'?" Well, today, guys, we're diving deep into the nitty-gritty of solving for a variable, specifically $c$, in the equation $61 = -9c + 7$. This isn't just about crunching numbers; it's about understanding the logic, the steps, and the satisfaction of finding that elusive value. So, buckle up, grab your metaphorical calculators, and let's break this down piece by piece. We're going to make sure you not only solve this equation but also understand why you're doing each step. Because let's be real, math is way cooler when you get it, right?

Understanding the Equation: What's $c$ Trying to Tell Us?

Alright, let's first get cozy with our equation: $61 = -9c + 7$. What does this even mean? Think of it like a puzzle. We have a known value on the left side, 61, and on the right side, we have a mix of numbers and a mystery variable, $c$. The equation is telling us that 61 is equal to the result of some operation involving $-9$ multiplied by $c$, and then 7 is added to that product. Our mission, should we choose to accept it, is to isolate $c$ – to get it all by itself on one side of the equation. Why? Because finding the value of $c$ is like finding the key to unlock the entire puzzle. It tells us the specific number that makes this statement true. It’s the answer that balances the scales. So, when we see $61 = -9c + 7$, we're not just looking at symbols; we're looking at a relationship between numbers, and our job is to figure out the missing piece of that relationship.

The Goal: Isolating $c$

Our primary objective when solving for $c$ in any equation is isolation. We want $c$ to be the star of the show, standing alone on one side of the equals sign. This means we need to peel away any numbers or operations that are attached to it. Think of it like unwrapping a present. You've got the main gift inside (that's our $c$), but it's covered in layers of wrapping paper and maybe even a bow (those are the $-9$ and the $+7$). We need to carefully remove each layer without damaging the gift. In algebraic terms, this means using inverse operations. For every operation being done to $c$, we perform the opposite operation to both sides of the equation. This is crucial because whatever you do to one side of an equation, you must do to the other to maintain the balance. If you don't, your equation becomes untrue, and your puzzle remains unsolved. So, our journey to solve for $c$ is essentially a journey of strategic unwrapping, applying opposite actions to keep things equal.

Step-by-Step: Unraveling the Equation

Now, let's get our hands dirty with the actual solving process for $61 = -9c + 7$. We're going to tackle this systematically, making sure every move is logical and keeps our equation balanced. Remember, the goal is to get $c$ by itself. We'll use the order of operations in reverse, often called the Reverse PEMDAS or SADMEP (Subtraction/Addition, Division/Multiplication, Exponents, Parentheses), to undo what's been done to $c$.

Step 1: Tackling the Constant Term

First off, let's look at the right side of the equation: $-9c + 7$. We see a term with $c$ ($-9c$) and a constant term ($+7$). Typically, we deal with the constant term first when isolating a variable. Our constant term is $+7$. To get rid of it, we need to perform the inverse operation. The inverse of adding 7 is subtracting 7. But here's the golden rule: whatever we do to one side, we must do to the other. So, we're going to subtract 7 from both sides of the equation.

Here’s how it looks:

$61 - 7 = -9c + 7 - 7$

Now, let's simplify both sides. On the left, $61 - 7$ gives us 54. On the right, the $+7$ and $-7$ cancel each other out, leaving us with just $-9c$.

Our equation has now transformed into:

$54 = -9c$

See? We've successfully removed the constant term from the side with $c$, bringing us one step closer to our goal of isolating $c$.

Step 2: Dealing with the Coefficient

We're looking good, guys! Our equation is now $54 = -9c$. Notice that $c$ is being multiplied by $-9$. This $-9$ is called the coefficient. To isolate $c$, we need to undo this multiplication. The inverse operation of multiplication is division. So, we need to divide both sides of the equation by $-9$.

Let's write that down:

rac{54}{-9} = rac{-9c}{-9}

Now, let's simplify. On the left side, $54 ext{ divided by } -9$ equals $-6$. On the right side, the $-9$ in the numerator and the $-9$ in the denominator cancel each other out, leaving us with just $c$.

And voilà! We have:

$-6 = c$

This tells us that $c$ is equal to $-6$. We've successfully isolated $c$ and found its value!

Verification: Is Our Answer Correct?

Okay, so we've crunched the numbers and arrived at $c = -6$. But are we sure? In math, there's a fantastic practice called verification or checking your work. It's like double-checking your homework before handing it in – it ensures accuracy and builds confidence. We can plug our found value of $c$ back into the original equation, $61 = -9c + 7$, and see if the equation holds true. If the left side equals the right side, then our solution is spot on!

Let's substitute $-6$ for $c$ in the original equation:

$61 = -9(-6) + 7$

First, we perform the multiplication: $-9 imes -6$. Remember, a negative number multiplied by a negative number results in a positive number. So, $-9 imes -6 = 54$.

Now, our equation looks like this:

$61 = 54 + 7$

Next, we perform the addition on the right side: $54 + 7 = 61$.

So, the equation becomes:

$61 = 61$

Boom! The left side (61) is indeed equal to the right side (61). This confirms that our solution, $c = -6$, is absolutely correct. High fives all around!

Conclusion: You've Solved for $c$!

So there you have it, folks! We've successfully navigated the steps to solve for $c$ in the equation $61 = -9c + 7$. We started by understanding the goal – isolating $c$ – and then systematically applied inverse operations. First, we subtracted 7 from both sides to get rid of the constant term, transforming the equation into $54 = -9c$. Then, we divided both sides by $-9$ to undo the multiplication and isolate $c$, leading us to the answer $c = -6$. Finally, we verified our answer by plugging it back into the original equation, confirming that 61 = 61. This process of isolating a variable is fundamental in algebra and opens the door to solving much more complex problems. Keep practicing these steps, and you'll be a math whiz in no time! Remember, every equation is just a puzzle waiting to be solved, and with a little logic and practice, you've got this!