Solving For X In 2(x-3)+9=3(x+1)+x A Step-by-Step Guide

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Hey guys! Ever stumbled upon an equation that looks like a jumbled mess of numbers and letters? Well, you're not alone! Today, we're going to dive deep into solving one such equation: 2(x-3)+9=3(x+1)+x. This equation might seem intimidating at first glance, but trust me, with a step-by-step approach and a little bit of algebraic know-how, we can crack this code and find the value of 'x'.

Unpacking the Equation: 2(x-3)+9=3(x+1)+x

Before we jump into solving, let's break down the equation. We have variables ('x'), constants (numbers), and operations (addition, subtraction, multiplication). The goal here is to isolate 'x' on one side of the equation to find its value. Think of it like a puzzle – we need to rearrange the pieces to reveal the solution. So, let’s embark on this mathematical adventure together, making sure we understand every twist and turn. Remember, mathematics isn't just about finding the answer; it's about understanding the process.

The Importance of Order of Operations

Before we even think about simplifying the equation, we need to talk about the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This is the golden rule of mathematics that dictates the sequence in which we perform operations. Ignoring this order can lead to incorrect solutions, and we definitely don’t want that. So, always keep PEMDAS in mind as our guiding star throughout this problem.

Step 1: Distribute Those Numbers!

The first thing we need to tackle are those parentheses. We're going to use the distributive property, which means multiplying the number outside the parentheses by each term inside. Let's start with the left side of the equation, where we have 2(x-3). We multiply 2 by 'x' and then by -3, which gives us 2x - 6. Now, let's do the same on the right side with 3(x+1). Multiplying 3 by 'x' and then by 1 gives us 3x + 3. Distributive property, you are our friend!

Step 2: Rewrite the Equation

Now that we've distributed, let's rewrite the equation with the simplified terms: 2x - 6 + 9 = 3x + 3 + x. Doesn't that look a little less intimidating? We've taken the first step in decluttering our equation and making it more manageable. This is a crucial step because it sets the stage for further simplification and ultimately helps us isolate 'x'. Always remember, simplification is key to solving complex equations.

Step 3: Combine Like Terms

Next up, we need to combine the like terms on each side of the equation. Like terms are those that have the same variable raised to the same power (or are just constants). On the left side, we have -6 and +9, which combine to give us +3. So, the left side becomes 2x + 3. On the right side, we have 3x and x, which combine to give us 4x. So, the right side becomes 4x + 3. Combining like terms is like tidying up a messy room – it makes everything easier to see and work with.

Step 4: Isolate the Variable Terms

Our next goal is to get all the 'x' terms on one side of the equation. To do this, we can subtract 2x from both sides. This keeps the equation balanced, which is super important! Subtracting 2x from both sides of 2x + 3 = 4x + 3 gives us 3 = 2x + 3. See how we're slowly but surely isolating 'x'? This is the magic of algebra at work.

Step 5: Isolate the Constant Terms

Now, let's isolate the constant terms. We want to get all the numbers without 'x' on one side of the equation. We can do this by subtracting 3 from both sides. Subtracting 3 from both sides of 3 = 2x + 3 gives us 0 = 2x. We're getting closer and closer to unlocking the value of 'x'! Each step we take brings us closer to the final answer.

Step 6: Solve for x

Finally, we're at the last step! To solve for 'x', we need to get it all by itself. We have 0 = 2x, which means 2 times 'x' equals 0. To undo the multiplication, we divide both sides by 2. Dividing both sides by 2 gives us x = 0. And there you have it! We've cracked the code and found the value of 'x'. Congratulations, you've conquered this algebraic challenge!

Checking Our Solution: The Importance of Verification

But wait, we're not done just yet! It's always a good idea to check our solution to make sure we didn't make any mistakes along the way. To do this, we substitute our value of x (which is 0) back into the original equation: 2(x-3)+9=3(x+1)+x. Let’s plug in x = 0 and see what happens.

The Verification Process

Substituting x = 0 into the original equation, we get: 2(0-3)+9=3(0+1)+0. Now, let's simplify both sides. On the left side, we have 2(-3) + 9, which equals -6 + 9, which equals 3. On the right side, we have 3(1) + 0, which equals 3. So, we have 3 = 3. This means our solution is correct! Checking our work is like having a safety net – it gives us the confidence that we've got the right answer.

Mastering Algebraic Equations: Tips and Tricks

Solving algebraic equations like this one is a fundamental skill in mathematics. But like any skill, it takes practice to master. Here are a few tips and tricks to help you on your journey:

  • Always follow the order of operations (PEMDAS). This is the foundation of algebraic manipulation.
  • Simplify both sides of the equation as much as possible before isolating the variable. This makes the equation easier to work with.
  • Remember to perform the same operation on both sides of the equation to maintain balance. Think of it like a seesaw – you need to keep it balanced to get the right answer.
  • Check your solution by substituting it back into the original equation. This helps you catch any errors.
  • Practice, practice, practice! The more you solve equations, the better you'll become at it.

Real-World Applications of Algebraic Equations

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