Solving For C Step-by-Step Guide

Hey everyone! Today, we're diving into a fundamental algebraic problem: solving for a variable. In this case, we're tackling the equation $1.9 - 2.2c = 3.88$ to find the value of 'c'. Don't worry if algebra feels a bit daunting – we'll break it down step-by-step so it's super clear and easy to follow. Think of it as a puzzle where we're trying to isolate 'c' on one side of the equation. By the end of this guide, you'll not only know how to solve this specific problem but also have a solid grasp of the general principles involved in solving linear equations. So, grab your pencils and let's get started!

Understanding the Equation

Before we jump into the solution, let's take a moment to really understand the equation we're dealing with: $1.9 - 2.2c = 3.88$. Equations, at their heart, are statements of balance. The equals sign (=) tells us that whatever is on the left side is exactly the same in value as what's on the right side. Our goal here is to figure out what value of 'c' will make this balance true. The left side of the equation features a constant term (1.9) and a term involving our variable 'c' (-2.2c). This '-2.2c' means '-2.2 multiplied by c'. Remember, in algebra, we often omit the multiplication symbol for brevity. The right side of the equation is simply a constant, 3.88. Now that we've dissected the equation, the roadmap to solving for 'c' becomes clearer. We need to manipulate the equation, using valid algebraic operations, to isolate 'c' on one side. This usually involves undoing the operations that are currently being applied to 'c'. Think of it like peeling away layers to reveal the treasure hidden inside – in this case, the value of 'c'!

Isolating the Variable Term

Our primary goal in solving for 'c' is to isolate it on one side of the equation. This means we want to get the term containing 'c' (which is -2.2c in our case) all by itself on one side. To do this, we need to get rid of the constant term that's on the same side – the 1.9. Remember, the key to maintaining balance in an equation is that whatever operation you perform on one side, you must perform the exact same operation on the other side. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level. So, how do we get rid of the 1.9? Since it's being added (even though there's a subtraction sign later, the 1.9 is positive), we need to do the opposite operation: subtraction. We'll subtract 1.9 from both sides of the equation. This gives us: $1.9 - 2.2c - 1.9 = 3.88 - 1.9$. On the left side, the 1.9 and -1.9 cancel each other out, leaving us with just -2.2c. On the right side, 3.88 - 1.9 equals 1.98. So, our equation now looks like this: $-2.2c = 1.98$. We've successfully isolated the variable term! We're one step closer to finding the value of 'c'.

Solving for 'c'

We've made excellent progress! We've isolated the term with 'c', and our equation now reads: $-2.2c = 1.98$. Remember, -2.2c means -2.2 multiplied by 'c'. To finally get 'c' by itself, we need to undo this multiplication. The opposite of multiplication is division, so we'll divide both sides of the equation by -2.2. This is crucial – we need to divide by the entire coefficient of 'c', including the negative sign. This gives us: $(-2.2c) / (-2.2) = 1.98 / (-2.2)$. On the left side, -2.2 divided by -2.2 is simply 1, leaving us with 1 * c, which is just 'c'. On the right side, 1.98 divided by -2.2 equals -0.9. Therefore, our solution is: $c = -0.9$. We've done it! We've successfully solved for 'c'. To double-check our answer, we can always substitute -0.9 back into the original equation and see if it holds true.

Verification

It's always a fantastic idea to verify your solution in algebra. It's like having a built-in error detector! To verify our solution, $c = -0.9$, we'll substitute it back into the original equation: $1.9 - 2.2c = 3.88$. Replacing 'c' with -0.9, we get: $1.9 - 2.2(-0.9) = 3.88$. Now, let's simplify. First, we perform the multiplication: -2.2 multiplied by -0.9 equals 1.98 (a negative times a negative is a positive!). So, our equation becomes: $1.9 + 1.98 = 3.88$. Next, we add 1.9 and 1.98, which indeed equals 3.88. Therefore, we have: $3.88 = 3.88$. This is a true statement! Since substituting our solution back into the original equation results in a true statement, we can be confident that our answer, $c = -0.9$, is correct. Verification is a powerful tool in algebra, and it's something you should always try to do, especially on tests or quizzes. It can save you from making simple mistakes and ensure you get the right answer.

Conclusion

Great job, guys! You've successfully solved for 'c' in the equation $1.9 - 2.2c = 3.88$. We walked through the process step-by-step, from understanding the equation and isolating the variable term to finally solving for 'c' and verifying our solution. Remember, the key to solving algebraic equations is to perform the same operations on both sides to maintain balance and to undo the operations that are being applied to the variable. This problem illustrates fundamental algebraic principles that you'll use again and again. So, if you feel a bit shaky on any of the steps, don't hesitate to review them. The more you practice, the more comfortable you'll become with algebra. Solving for variables is a crucial skill in math and science, so mastering it now will pay off big time in the future. Keep practicing, keep asking questions, and you'll become an algebra ace in no time!