Simplify Expressions: A Step-by-Step Guide

Hey guys! Ever feel like algebraic expressions are just a jumbled mess of numbers and letters? Don't worry, you're not alone! But the truth is, simplifying these expressions can be super satisfying, like cracking a secret code. In this article, we're going to break down a specific problem step-by-step, so you can master this skill too. Let's dive in!

The Challenge: Untangling the Expression

Our mission, should we choose to accept it, is to simplify the following expression:

$$-2x^2 + 8x - 9 + 4x + 7x^2 + 2$$

At first glance, it might look a bit intimidating, but fear not! We're going to use a simple yet powerful technique called combining like terms. This is the key to making sense of these expressions. So, what exactly are "like terms"? Well, they're the terms that have the same variable raised to the same power. Think of them as belonging to the same family. For example, $-2x^2$ and $7x^2$ are like terms because they both have $x^2$. Similarly, $8x$ and $4x$ are like terms because they both have $x$, and the constants $-9$ and $2$ are like terms because they're just plain numbers.

Step 1: Spotting the Like Terms

The first step in simplifying our expression is to identify all the like terms. Let's rewrite the expression and group the like terms together to make it visually clearer:

$$(-2x^2 + 7x^2) + (8x + 4x) + (-9 + 2)$$

See how we've grouped the $x^2$ terms, the $x$ terms, and the constants? This makes it much easier to see what we need to combine.

Step 2: Combining the Coefficients

Now comes the fun part: combining the like terms! To do this, we simply add or subtract the coefficients (the numbers in front of the variables) of the like terms. Let's start with the $x^2$ terms:

$$-2x^2 + 7x^2 = (7 - 2)x^2 = 5x^2$$

So, we have $5x^2$. Next, let's combine the $x$ terms:

$$8x + 4x = (8 + 4)x = 12x$$

Now we have $12x$. Finally, let's combine the constants:

$$-9 + 2 = -7$$

And we're left with $-7$.

Step 3: Putting It All Together

We've successfully combined all the like terms! Now, let's put everything together to get our simplified expression:

$$5x^2 + 12x - 7$$

And there you have it! Our original expression $-2x^2 + 8x - 9 + 4x + 7x^2 + 2$ simplifies to $5x^2 + 12x - 7$.

Checking Our Answer

It's always a good idea to double-check our work to make sure we haven't made any mistakes. One way to do this is to substitute a value for $x$ in both the original expression and the simplified expression. If we get the same result, we can be pretty confident that we've simplified correctly. Let's try substituting $x = 1$:

Original expression:

$$-2(1)^2 + 8(1) - 9 + 4(1) + 7(1)^2 + 2 = -2 + 8 - 9 + 4 + 7 + 2 = 10$$

Simplified expression:

$$5(1)^2 + 12(1) - 7 = 5 + 12 - 7 = 10$$

We got the same result! That's a good sign. Let's try another value, just to be sure. Let's use $x = 0$:

Original expression:

$$-2(0)^2 + 8(0) - 9 + 4(0) + 7(0)^2 + 2 = -9 + 2 = -7$$

Simplified expression:

$$5(0)^2 + 12(0) - 7 = -7$$

Again, we got the same result. We're feeling pretty good about our answer!

The Answer: Cracking the Code

So, which of the given options matches our simplified expression? Let's take a look:

• A. $-5x^2 + 4x + 11$ (Nope!)
• B. $-9x^2 + 4x - 7$ (Definitely not!)
• C. $-9x^2 - 13x + 11$ (Not even close!)
• D. $5x^2 + 12x - 7$ (Bingo!)

Option D is the correct answer! We've successfully simplified the expression and found the equivalent form.

Why Combining Like Terms Matters

Combining like terms isn't just a random math trick; it's a fundamental skill that's used throughout algebra and beyond. It's essential for solving equations, graphing functions, and simplifying more complex expressions. Think of it as the foundation upon which many other mathematical concepts are built.

For example, imagine you're solving an equation like this:

$$3x + 5 - x = 9$$

Before you can isolate $x$, you need to combine the like terms on the left side of the equation:

$$(3x - x) + 5 = 9$$

$$2x + 5 = 9$$

Now the equation is much simpler to solve. See how important combining like terms is?

Tips and Tricks for Mastering Like Terms

Here are a few tips to help you become a pro at combining like terms:

• Highlight or underline like terms: This can help you visually group them and avoid mistakes.
• Pay attention to the signs: Remember to include the signs (positive or negative) in front of the terms when combining them.
• Write out the steps: Don't try to do everything in your head. Writing out each step can help you stay organized and catch errors.
• Practice, practice, practice! The more you practice, the easier it will become.

Practice Problems: Test Your Skills

Ready to put your skills to the test? Try simplifying these expressions:

1. $$4y^2 - 7y + 2 + 3y - 5y^2 - 1$$

2. $$9a - 2b + 5a + 6b - 3$$

3. $$-6z^3 + 2z - 8 + 4z^3 - z + 10$$

(Answers are at the end of the article, so don't peek just yet!)

Beyond the Basics: Real-World Applications

You might be wondering, "When am I ever going to use this in real life?" Well, the truth is, simplifying algebraic expressions has many practical applications. It's used in fields like:

• Engineering: Engineers use algebraic expressions to model and analyze systems, such as electrical circuits and mechanical structures.
• Computer science: Programmers use algebraic expressions to write algorithms and solve problems.
• Economics: Economists use algebraic expressions to model economic trends and make predictions.
• Finance: Financial analysts use algebraic expressions to calculate investments and manage risk.

Even in everyday life, you might use the principles of combining like terms without even realizing it. For example, if you're calculating the total cost of items at a store, you're essentially combining like terms (the prices of the items).

Conclusion: You've Got This!

Simplifying algebraic expressions might seem tricky at first, but with a little practice, you can master it. Remember the key: identify the like terms, combine their coefficients, and double-check your work. By following these steps, you'll be simplifying expressions like a pro in no time! And remember, understanding this concept is crucial for success in higher-level math courses and in many real-world applications.

So, keep practicing, keep exploring, and never stop learning! You've got this!

(Answers to practice problems:

1. $$-y^2 - 4y + 1$$

2. $$14a + 4b - 3$$

3. -2z^3 + z + 2$)