Solving The Equation: 10 + 5 = (18p - 6) / 2

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Solving the Equation: 10 + 5 = (18p - 6) / 2

Hey guys! Today, we're diving into a fun little math problem. We're going to break down how to solve the equation 10 + 5 = (18p - 6) / 2. Don’t worry, it’s not as scary as it looks! We’ll take it step by step so you can follow along easily. Whether you're brushing up on your algebra skills or just curious, this guide will help you understand the process. So, grab your pencils, and let's get started!

Understanding the Basics

Before we jump into solving, let’s make sure we understand the basic principles at play here. This equation involves simple arithmetic, a variable (p), and a fraction. Our goal is to isolate p on one side of the equation to find its value. Remember, whatever we do to one side of the equation, we must do to the other to keep it balanced. This is the golden rule of equation solving!

The Order of Operations

First things first, let’s talk about the order of operations. You might have heard of PEMDAS or BODMAS. It stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order helps us simplify expressions correctly. In our equation, we'll deal with the division part first, but we also need to consider the terms inside the parentheses.

Isolating the Variable

The main aim when solving equations is to isolate the variable. In our case, we want p to be all by itself on one side of the equals sign. To do this, we'll use inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and so are multiplication and division. By strategically using inverse operations, we can peel away the layers around p until it’s all alone.

Step-by-Step Solution

Okay, let’s get down to business and solve this equation step by step. I'll break it down so it's super clear.

Step 1: Simplify Both Sides

Our equation is 10 + 5 = (18p - 6) / 2. The first thing we can do is simplify the left side. 10 + 5 is simply 15. So our equation now looks like this:

15 = (18p - 6) / 2

Step 2: Eliminate the Fraction

Fractions can sometimes make things look complicated, but they're easy to handle. To get rid of the fraction, we need to multiply both sides of the equation by the denominator, which is 2 in this case. This is because multiplying by 2 will cancel out the division by 2 on the right side.

So, we multiply both sides by 2:

2 * 15 = 2 * [(18p - 6) / 2]

This simplifies to:

30 = 18p - 6

Step 3: Isolate the Term with 'p'

Now, we want to get the term with p (which is 18p) by itself on one side. To do this, we need to get rid of the -6. The inverse operation of subtraction is addition, so we'll add 6 to both sides of the equation:

30 + 6 = 18p - 6 + 6

This simplifies to:

36 = 18p

Step 4: Solve for 'p'

We're almost there! Now we have 36 = 18p. To solve for p, we need to get it completely alone. The 18 is multiplying p, so the inverse operation we need is division. We'll divide both sides of the equation by 18:

36 / 18 = 18p / 18

This simplifies to:

2 = p

So, the solution to the equation is p = 2!

Checking Our Work

It's always a good idea to check your answer to make sure you didn't make any mistakes. To do this, we'll substitute p = 2 back into the original equation:

10 + 5 = (18 * 2 - 6) / 2

Let's simplify:

15 = (36 - 6) / 2

15 = 30 / 2

15 = 15

Since both sides of the equation are equal, our solution p = 2 is correct!

Common Mistakes to Avoid

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

  • Forgetting the Order of Operations: Always remember PEMDAS/BODMAS. Simplify inside parentheses first, then deal with exponents, multiplication and division, and finally addition and subtraction.
  • Not Applying Operations to Both Sides: Whatever you do to one side of the equation, you must do to the other. If you add 6 to one side, make sure you add 6 to the other side too.
  • Incorrectly Distributing: If you have a number multiplying a group of terms in parentheses, make sure you distribute it to each term. For example, if you have 2(x + 3), it becomes 2x + 6, not 2x + 3.
  • Arithmetic Errors: Simple addition, subtraction, multiplication, or division errors can throw off your entire solution. Double-check your work, especially if you're working quickly.

Practice Problems

Want to become a pro at solving equations? The best way is to practice! Here are a few similar problems you can try:

  1. 5 + 3 = (4x - 2) / 2
  2. 20 + 10 = (6y + 12) / 3
  3. 8 + 2 = (5z - 5) / 5

Try solving these problems on your own, and then check your answers. The more you practice, the more comfortable you'll become with solving equations.

Real-World Applications

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