Solving For 'y': A Step-by-Step Guide

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Hey guys! Let's dive into the world of algebra and learn how to solve for a variable, specifically 'y', in an equation. We'll be working with the equation: C = 6(y + 6). Don't worry if this looks a bit intimidating at first – we'll break it down into easy, digestible steps. Solving for a variable means isolating it on one side of the equation, so we have an expression equal to y. This is a fundamental skill in math and is super useful in all sorts of real-world scenarios, from calculating the cost of a purchase to figuring out the best route for a road trip. The key is to understand the order of operations and how to use inverse operations to get 'y' by itself. We'll go through the process step by step, explaining each action along the way, ensuring that anyone can follow along. Ready? Let's do this!

Understanding the Basics: The Equation's Anatomy

First, let's understand what we're dealing with. The equation C = 6(y + 6) is an algebraic equation. It includes a variable (y), constants (6 and 6), and the operation of multiplication and addition. The goal is to isolate y. Think of an equation like a balanced scale; to keep it balanced, any operation you perform on one side of the equation must also be performed on the other side. Let’s identify the components to better understand the question.

  • C: This can represent a constant, or a value. The variable, 'y' is the one we want to solve for.
  • 6: This is the coefficient, the number that is multiplied by the expression within the parentheses. The parenthesis shows that we need to multiply 6 to all values within the parentheses.
  • (y + 6): This is an expression. The variable y is added to the constant number 6.

Before we start, remember the order of operations (PEMDAS/BODMAS) to help us out! This is the order in which we solve equations; Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). With this in mind, let’s begin solving the equation. Remember, always keep the equation balanced.

Step-by-Step Solution: Unveiling the Value of 'y'

Alright, let's solve for y in the equation C = 6(y + 6). Here’s how we do it, step-by-step:

  1. Distribute the 6: The first step is to get rid of those parentheses by distributing the 6 across the terms inside. This means multiplying the 6 by both y and 6.

    • 6 * y = 6y
    • 6 * 6 = 36
    • So, our equation now becomes: C = 6y + 36.

  2. Isolate the Term with 'y': Our next goal is to get the term with 'y' (which is 6y) by itself on one side of the equation. To do this, we need to get rid of the +36. We can do this by subtracting 36 from both sides of the equation. Remember, we need to do the same thing on both sides to keep the equation balanced.

    • C - 36 = 6y + 36 - 36
    • This simplifies to: C - 36 = 6y.

  3. Isolate 'y': Now we want to get 'y' alone. 'y' is currently being multiplied by 6. To get rid of the multiplication, we need to do the opposite: divide. Divide both sides of the equation by 6.

    • (C - 36) / 6 = (6y) / 6
    • This simplifies to: (C - 36) / 6 = y. Or we can rewrite this as: y = (C - 36) / 6.

And there you have it, folks! We've solved for y. The solution is y = (C - 36) / 6. We have successfully isolated 'y'. The value of 'y' depends on the value of 'C'.

Verification and Further Exploration

To ensure our work is correct, we can verify our solution. Although we cannot plug in a specific value to make sure that it's correct because C is a variable, we can explain the logic on how to verify it. If you were provided a value for C, all you have to do is plug that value into both sides of the original equation. Then simplify to see if both sides are equal, which confirms your solution. If the sides are equal, then your solution is correct.

Let's say, for example, C = 72.

  1. Substitute: 72 = 6(y + 6)
  2. Isolate (y + 6) by dividing both sides by 6:
    • 72 / 6 = (6(y + 6)) / 6
    • 12 = y + 6
  3. Isolate y by subtracting 6 from both sides:
    • 12 - 6 = y + 6 - 6
    • 6 = y
    • y = 6

Now, let's verify if the solution is correct.

  1. Original equation: C = 6(y + 6)
  2. Substitute the values, C = 72, and y = 6:
    • 72 = 6(6 + 6)
    • 72 = 6(12)
    • 72 = 72

Since both sides are equal, our solution is correct. We have successfully solved for 'y'. This skill is fundamental to solving more complex algebraic problems. Keep practicing, and you'll become a pro in no time! Keep in mind that we can solve equations like this with various variables and complexities. With more practice, it will be easier and easier to solve for the value you're looking for.

Tips and Tricks for Solving Equations

Solving for y can seem tricky, but with the right approach, it becomes a breeze. Here are some extra tips to help you along the way.

  1. Master the Order of Operations: Always, always, always follow the order of operations (PEMDAS/BODMAS). This is the golden rule!
  2. Practice Makes Perfect: The more equations you solve, the better you'll get. Try different types of equations, change the values, and solve them again. The process will be ingrained in your brain.
  3. Double-Check Your Work: Mistakes happen, so it’s always a good idea to double-check your steps, especially when you are a beginner. Substitute your solution back into the original equation. If the equation holds true, you're golden!
  4. Break it Down: If an equation looks overwhelming, break it down into smaller, manageable steps. This will make the entire process less daunting.
  5. Use Visual Aids: Drawing diagrams, using color-coding, or writing out each step can help you stay organized and avoid mistakes.

Conclusion: Your Journey into Algebra

So there you have it, my friends! We’ve successfully solved for y in the equation C = 6(y + 6). We started with the basic equation, applied the order of operations, and used inverse operations to isolate y. Remember that solving for a variable is a fundamental skill in algebra and is used extensively in a variety of fields. The key is to practice, stay organized, and don’t be afraid to ask for help when you need it. Keep practicing, and you'll find that solving equations becomes easier and more intuitive. Keep learning and growing, and you'll see how useful this skill can be. You got this!