Solving For R: A Step-by-Step Guide To The Equation

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Solving for r: A Step-by-Step Guide to the Equation

Hey guys! Today, let's dive into solving a cool equation for 'r'. It might look a bit intimidating at first, but trust me, we'll break it down into simple steps so everyone can follow along. Our mission is to isolate 'r' and figure out what value makes the equation true. So, grab your thinking caps, and let's get started!

Understanding the Equation

First things first, let's take a good look at our equation: 0 = 19 + 12 + 3(8 - r) + 2r + 4. Before we start moving things around, it's super important to understand what each part means. We've got constants (numbers that don't change), variables (that's our 'r'), and some parentheses that tell us the order in which we need to do things. Remember PEMDAS/BODMAS? (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). That's our guide here. Understanding the equation is like having a map before a treasure hunt; it tells us where we're going and how to get there.

Step-by-Step Solution

Step 1: Simplify the Constants

Okay, let's start by simplifying all the constant numbers. We have 19, 12, and 4 that we can add together. So, 19 + 12 + 4 equals 35. Now our equation looks a little cleaner: 0 = 35 + 3(8 - r) + 2r. See? We're already making progress! Simplifying constants is like clearing the clutter on your desk before starting a big project; it makes everything less confusing.

Step 2: Distribute

Next up, we need to deal with those parentheses. We have 3(8 - r). This means we need to multiply 3 by both 8 and -r. So, 3 times 8 is 24, and 3 times -r is -3r. Now our equation looks like this: 0 = 35 + 24 - 3r + 2r. Distributing is like unpacking a box of goodies; you're revealing the contents and making them accessible.

Step 3: Combine Like Terms

Now, let's combine all the like terms. We can add the constants 35 and 24, which gives us 59. And we can combine the 'r' terms: -3r + 2r, which equals -r. So, our equation now reads: 0 = 59 - r. Combining like terms is like sorting your socks; you're putting similar things together to make them easier to manage.

Step 4: Isolate 'r'

Our goal is to get 'r' all by itself on one side of the equation. To do this, we can add 'r' to both sides of the equation. This gives us: r = 59. And there you have it! We've solved for 'r'. Isolating 'r' is like finding the missing piece of a puzzle; once you have it, everything else falls into place.

Verification

To make sure we didn't make any mistakes along the way, let's plug our solution, r = 59, back into the original equation and see if it holds true.

Original equation: 0 = 19 + 12 + 3(8 - r) + 2r + 4

Substitute r = 59: 0 = 19 + 12 + 3(8 - 59) + 2(59) + 4

Simplify: 0 = 19 + 12 + 3(-51) + 118 + 4

Continue simplifying: 0 = 19 + 12 - 153 + 118 + 4

Combine the numbers: 0 = 153 - 153

Final result: 0 = 0

Since the equation holds true, our solution r = 59 is correct. Verification is like double-checking your work; it ensures that you've arrived at the right answer.

Common Mistakes to Avoid

When solving equations, it's easy to make a few common mistakes. Here are some things to watch out for:

  1. Forgetting the Order of Operations: Always remember PEMDAS/BODMAS. Do parentheses first, then exponents, then multiplication and division, and finally addition and subtraction.
  2. Distribution Errors: Make sure you multiply the number outside the parentheses by every term inside the parentheses. For example, in 3(8 - r), you need to multiply 3 by both 8 and -r.
  3. Combining Unlike Terms: You can only combine terms that have the same variable. For example, you can combine -3r and 2r, but you can't combine -3r and 35.
  4. Sign Errors: Pay close attention to the signs (positive and negative) when you're adding, subtracting, multiplying, and dividing. A small sign error can throw off your entire answer.
  5. Not Verifying Your Solution: Always plug your solution back into the original equation to make sure it works. This is the best way to catch any mistakes you might have made.

Avoiding these common mistakes is like wearing the right gear for a hike; it helps you navigate the terrain more safely and avoid unnecessary pitfalls.

Practice Problems

Want to become a pro at solving equations? Here are a few practice problems you can try:

  1. Solve for x: 2x + 5 = 15
  2. Solve for y: 3(y - 2) = 9
  3. Solve for z: 4z - 7 = 2z + 1
  4. Solve for a: 5a + 3 = 2a - 6
  5. Solve for b: -2(b + 4) = 10

Solving these practice problems is like practicing your scales on the piano; the more you do it, the better you'll become.

Conclusion

So there you have it! We've successfully solved for 'r' in the equation 0 = 19 + 12 + 3(8 - r) + 2r + 4. Remember, solving equations is all about breaking them down into manageable steps, understanding the rules, and avoiding common mistakes. Keep practicing, and you'll become a master in no time!

Remember, math isn't just about numbers and equations; it's about problem-solving, critical thinking, and logical reasoning. These skills are valuable in all aspects of life, so keep challenging yourself and never stop learning.

Keep up the great work, and I'll catch you in the next math adventure!