Solving Equations Step By Step What Is The Solution For 9(w-4)-7w=5(3w-2)

Hey there, math enthusiasts! Today, we're diving headfirst into the exciting world of algebra to tackle a question that might seem a bit daunting at first glance: What is the solution to the equation 9(w-4) - 7w = 5(3w-2)? Don't worry, though! We're going to break it down step by step, making sure everyone, from math novices to seasoned pros, can follow along and understand the process. So, grab your pencils, notebooks, and let's get started on this mathematical journey together!

Deciphering the Equation: A Step-by-Step Guide

The Initial Setup

Our mission, should we choose to accept it (and we do!), is to find the value of 'w' that makes this equation true. The equation we're dealing with is 9(w-4) - 7w = 5(3w-2). It looks a bit tangled right now, but fear not! We're going to use some algebraic magic to simplify it. The key here is to remember the order of operations (PEMDAS/BODMAS) and to apply the distributive property correctly.

Step 1: Distributive Property – Spreading the Love

The first thing we need to do is get rid of those parentheses. To do this, we'll use the distributive property. This property tells us that a(b + c) = ab + ac. Basically, we need to multiply the number outside the parentheses by each term inside the parentheses.

Let's apply this to our equation:

• 9(w - 4) becomes 9 * w - 9 * 4, which simplifies to 9w - 36
• 5(3w - 2) becomes 5 * 3w - 5 * 2, which simplifies to 15w - 10

Now our equation looks like this: 9w - 36 - 7w = 15w - 10. See? Already, it's looking a bit cleaner!

Step 2: Combining Like Terms – Gathering the Family

Next up, we want to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power (or no variable at all, in the case of constants). On the left side of the equation, we have 9w and -7w, which are like terms. We also have the constant -36.

Let's combine those 'w' terms: 9w - 7w = 2w

So, the left side of the equation simplifies to 2w - 36.

Our equation now looks like this: 2w - 36 = 15w - 10. We're making progress!

Step 3: Isolating the Variable – Getting 'w' Alone

Our goal is to get 'w' by itself on one side of the equation. To do this, we need to move all the 'w' terms to one side and all the constant terms to the other side. Let's start by moving the 'w' terms. We have 2w on the left and 15w on the right. To keep things positive (because who doesn't love positivity?), let's move the 2w to the right side.

To do this, we subtract 2w from both sides of the equation:

2w - 36 - 2w = 15w - 10 - 2w

This simplifies to -36 = 13w - 10.

Now, let's move the constant terms. We have -36 on the left and -10 on the right. Let's move the -10 to the left side by adding 10 to both sides:

-36 + 10 = 13w - 10 + 10

This simplifies to -26 = 13w.

Step 4: Solving for 'w' – The Grand Finale

We're in the home stretch now! We have -26 = 13w. To solve for 'w', we need to get rid of the 13 that's multiplying it. We do this by dividing both sides of the equation by 13:

-26 / 13 = 13w / 13

This simplifies to -2 = w.

And there we have it! The solution to the equation is w = -2.

Verification: Did We Get It Right?

It's always a good idea to check our work, just to be sure we didn't make any sneaky mistakes along the way. To do this, we'll plug our solution, w = -2, back into the original equation and see if it holds true.

Original equation: 9(w - 4) - 7w = 5(3w - 2)

Substitute w = -2: 9(-2 - 4) - 7(-2) = 5(3(-2) - 2)

Now, let's simplify:

• 9(-6) - (-14) = 5(-6 - 2)
• -54 + 14 = 5(-8)
• -40 = -40

It checks out! Both sides of the equation are equal, so we know that w = -2 is indeed the correct solution.

Common Pitfalls and How to Avoid Them

Solving equations like this can be tricky, and it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

• Distributive Property Errors: Forgetting to distribute to all terms inside the parentheses or making mistakes with signs (e.g., multiplying a negative number). Solution: Double-check your work and make sure you're distributing correctly.
• Combining Unlike Terms: Trying to add or subtract terms that aren't like terms (e.g., adding 2w and -36). Solution: Only combine terms that have the same variable raised to the same power.
• Sign Errors: Making mistakes with positive and negative signs, especially when moving terms across the equals sign. Solution: Be extra careful with signs and double-check your work.
• Order of Operations: Not following the correct order of operations (PEMDAS/BODMAS). Solution: Always remember to do parentheses/brackets first, then exponents/orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

By being aware of these common mistakes and taking your time to work through each step carefully, you can avoid them and solve equations with confidence.

Real-World Applications: Where Does This Math Come in Handy?

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