Simplifying Algebraic Equations: A Step-by-Step Guide
Hey guys! Today, we're diving deep into the awesome world of algebra, specifically tackling a problem that might look a little intimidating at first glance: 5(x+5)-3(x+5)=3(x-2). Don't worry, though! By the end of this, you'll see how to break down these kinds of equations and solve for 'x' like a pro. We're going to go through this step-by-step, making sure you understand every move we make. So, grab your notebooks, maybe a snack, and let's get this mathematical party started!
Understanding the Basics: What's an Algebraic Equation?
Before we jump into solving our specific equation, let's quickly recap what an algebraic equation is. At its core, it's like a balanced scale. You have an expression on one side, an expression on the other, and an equals sign (=) in the middle, showing that both sides are equivalent. Our goal when solving an algebraic equation is to isolate the variable – usually 'x' – on one side of the equation. This means getting 'x' all by itself, so we can figure out its value. Think of it like a puzzle where you need to find the missing piece. In our case, the puzzle is 5(x+5)-3(x+5)=3(x-2), and the missing piece is the value of 'x' that makes this equation true.
Algebraic equations are super useful in real life, even if you don't always see them. They're used in everything from engineering and finance to figuring out how much paint you need for a room. The principles we're using today are fundamental, so mastering them will set you up for more complex math down the line. We'll be using properties like the distributive property and the order of operations (PEMDAS/BODMAS) to simplify and solve. It's all about applying these rules consistently and carefully. So, let's get ready to flex those math muscles!
Step 1: Simplify Both Sides of the Equation
Alright, team, let's tackle the first big step in solving 5(x+5)-3(x+5)=3(x-2): simplifying both sides. This is where we use our trusty distributive property. Remember, the distributive property is like saying you're sharing whatever is outside the parentheses with everything inside. So, for the left side, 5(x+5)-3(x+5), we have two terms being multiplied by parentheses. Let's break it down:
- First term: 5(x+5). Here, we multiply 5 by 'x' and then 5 by 5. This gives us
5*x + 5*5, which simplifies to5x + 25. - Second term: -3(x+5). This is crucial! Don't forget the negative sign in front of the 3. We multiply -3 by 'x' and then -3 by 5. This gives us
-3*x + (-3)*5, which simplifies to-3x - 15.
Now, we combine these simplified terms back on the left side: (5x + 25) + (-3x - 15). We can group the 'x' terms together and the constant terms together: (5x - 3x) + (25 - 15). Performing the arithmetic, we get 2x + 10.
See? The entire left side of our equation, 5(x+5)-3(x+5), has been neatly simplified to 2x + 10. This is a huge win!
Now, let's turn our attention to the right side of the equation: 3(x-2). Again, we use the distributive property. We multiply 3 by 'x' and then 3 by -2. This gives us 3*x + 3*(-2), which simplifies to 3x - 6.
So, after simplifying both sides, our original equation 5(x+5)-3(x+5)=3(x-2) transforms into a much cleaner form: 2x + 10 = 3x - 6. Isn't that much easier to look at? This simplification step is absolutely vital. If you skip this or make a mistake here, the rest of your solution will be off. So, always double-check your distribution and your arithmetic. Take your time, and be meticulous. You've got this!
Step 2: Combine Like Terms and Isolate the Variable
Now that we have our simplified equation, 2x + 10 = 3x - 6, it's time for the next phase: isolating the variable 'x'. This means getting all the 'x' terms on one side of the equation and all the constant numbers on the other side. It's like organizing your room – you want all your clothes in the closet and all your books on the shelf.
Let's start by moving the 'x' terms. We have 2x on the left and 3x on the right. It's generally easier to move the smaller 'x' term to avoid dealing with negative coefficients, but you can move either one. Let's subtract 2x from both sides of the equation to keep it balanced. Remember, whatever you do to one side, you must do to the other.
So, we have:
2x + 10 - 2x = 3x - 6 - 2x
This simplifies to:
10 = (3x - 2x) - 6
10 = x - 6
Look at that! We've successfully moved all the 'x' terms to the right side. Now, we need to move the constant terms. We have 10 on the left and -6 on the right. We want to get the x by itself, so we need to get rid of that -6. To do this, we'll do the opposite operation: add 6 to both sides of the equation.
10 + 6 = x - 6 + 6
This simplifies to:
16 = x
And there you have it! We've isolated 'x'. The solution to our equation is x = 16. This is a super satisfying step because you can see the variable taking shape and eventually standing alone. It requires a good understanding of inverse operations – addition undoes subtraction, subtraction undoes addition, multiplication undoes division, and vice versa. By applying these inverse operations systematically to both sides, we maintain the equality of the equation while moving terms around to achieve our goal of isolating 'x'. Don't be afraid to take your time with these steps; it's better to be slow and correct than fast and wrong!
Step 3: Verification - Checking Your Answer
So, we've solved 5(x+5)-3(x+5)=3(x-2) and found that x = 16. But how do we know for sure that we're right? This is where the verification step comes in, and trust me, guys, it's your best friend in algebra. It's like double-checking your work before submitting a big project. We take our solution, x = 16, and plug it back into the original equation to see if both sides are equal.
Our original equation was: 5(x+5)-3(x+5)=3(x-2)
Let's substitute x = 16 into the left side:
5(16 + 5) - 3(16 + 5)
First, calculate the values inside the parentheses:
16 + 5 = 21
Now, substitute that back in:
5(21) - 3(21)
Perform the multiplication:
5 * 21 = 105
3 * 21 = 63
Now, subtract:
105 - 63 = 42
So, the left side of the equation equals 42 when x = 16.
Now, let's substitute x = 16 into the right side:
3(x - 2)
3(16 - 2)
Calculate the value inside the parentheses:
16 - 2 = 14
Now, substitute that back in:
3(14)
Perform the multiplication:
3 * 14 = 42
So, the right side of the equation also equals 42 when x = 16.
Since the left side (42) equals the right side (42), our solution x = 16 is correct! This verification process is super important. It not only confirms your answer but also helps you catch any errors you might have made during the simplification or isolation steps. If you find that the two sides don't match, don't get discouraged! Just go back through your steps, check your arithmetic and your application of algebraic rules, and find where the mistake happened. It's all part of the learning process, and every mistake is a chance to get better. So, always, always verify your solutions!
Common Pitfalls and How to Avoid Them
Even with the best intentions, guys, we can sometimes stumble when solving algebraic equations. Let's talk about some common pitfalls and how you can steer clear of them when working with problems like 5(x+5)-3(x+5)=3(x-2).
1. Sign Errors (The Sneaky Culprits!)
This is probably the most common mistake. Forgetting a negative sign, especially when distributing, can completely change your answer. In our example, when we distributed the -3 in -3(x+5), we needed to multiply -3 by x (giving -3x) AND -3 by 5 (giving -15). If you just multiplied -3 by x and forgot the -15, your entire left side would be wrong. Tip: Always pay extra attention to the signs, especially when you have subtraction or negative numbers involved. It's helpful to circle or highlight negative signs before you start distributing.
2. Distributive Property Mishaps
This ties into sign errors but is broader. The distributive property means multiplying the term outside the parentheses by every term inside. Forgetting to distribute to the second term (or any subsequent terms) is a frequent slip-up. For instance, in 3(x-2), you must multiply 3 by x AND 3 by -2. Tip: Draw little arrows from the number outside the parentheses to each number inside to remind yourself to multiply by all of them.
3. Errors in Combining Like Terms
When simplifying, like combining 5x and -3x, or 25 and -15, make sure you're adding or subtracting correctly. 5x - 3x is 2x, but 5x - 7x is -2x. Similarly, 25 - 15 is 10, but 25 - 30 is -5. Tip: If you're unsure about adding integers, write out a number line or use a calculator for confirmation. Grouping like terms visually (e.g., circling all 'x' terms, boxing all constant terms) can also help.
4. Mistakes When Isolating the Variable
This is where inverse operations come into play. If you have +10 on one side and want to move it, you subtract 10. If you have -6 and want to move it, you add 6. If you have 2x and want to isolate x, you divide by 2. Trying to do too much at once can lead to errors. Tip: Perform only one operation per step. First, move all the variable terms to one side. Then, on a separate step, move all the constant terms to the other side. This systematic approach minimizes confusion.
5. Not Verifying Your Answer
As we saw in Step 3, verification is crucial! Skipping this step means you might submit an incorrect answer without even knowing it. Tip: Always, always plug your answer back into the original equation. It's the ultimate reality check. Even if you're short on time, take a minute or two to do this. It can save you from losing points on homework or tests.
By being aware of these common mistakes and employing the tips provided, you'll significantly increase your accuracy and confidence when solving algebraic equations. Remember, practice makes perfect, and every equation you solve is a step towards mastery!
Conclusion: Mastering Algebraic Equations
So there you have it, guys! We've successfully navigated the equation 5(x+5)-3(x+5)=3(x-2) from start to finish. We learned how to simplify complex expressions using the distributive property, how to combine like terms, and how to expertly isolate the variable 'x' using inverse operations. Most importantly, we emphasized the critical step of verification, ensuring our solution x = 16 was indeed correct by plugging it back into the original equation.
Solving algebraic equations might seem daunting at first, but by breaking them down into manageable steps and understanding the fundamental rules of algebra, you can tackle even the most challenging problems. Remember the power of the distributive property, the importance of keeping your equation balanced by performing the same operation on both sides, and the undeniable value of a final verification step. Don't be afraid of mistakes; they are simply stepping stones on the path to understanding. Keep practicing, stay curious, and you'll find yourself becoming more and more comfortable and confident with algebra. Keep up the great work, and happy solving!