Simplify -(3x-15)+2(3x+7): A Step-by-Step Guide

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Hey guys! Let's dive into simplifying this algebraic expression. It might look a bit daunting at first, but trust me, we'll break it down step-by-step, and you'll see it's totally manageable. Our mission today is to simplify $-(3x-15)+2(3x+7)$. This involves using the distributive property and combining like terms. So, grab your pencils, and let's get started!

Breaking Down the Expression

When you first glance at the expression $-(3x-15)+2(3x+7)$, you'll notice there are a couple of key operations we need to tackle: distribution and combining like terms. The distributive property is our best friend here. It allows us to multiply a single term by multiple terms inside parentheses. Think of it as spreading the love (or the multiplication, in this case) to everyone inside the brackets.

Let's start with the first part of the expression: $-(3x-15)$. The negative sign in front of the parentheses can be thought of as multiplying the entire expression inside by -1. So, we distribute the -1 across both terms inside the parentheses:

-1 * (3x) = -3x -1 * (-15) = +15

So, $-(3x-15)$ simplifies to $-3x + 15$. See? Not so scary when we break it down.

Now, let’s move on to the second part of the expression: $2(3x+7)$. Here, we need to distribute the 2 across both terms inside the parentheses:

2 * (3x) = 6x 2 * (7) = 14

So, $2(3x+7)$ simplifies to $6x + 14$. We're making progress, guys!

Combining Like Terms

Now that we've distributed, our expression looks like this: $-3x + 15 + 6x + 14$. The next step in simplifying is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have two terms with 'x' and two constant terms (numbers without variables).

The terms with 'x' are -3x and 6x. Let's combine them:

-3x + 6x = 3x

So, when we combine -3x and 6x, we get 3x. That's a nice, clean term.

Next, let’s combine the constant terms: 15 and 14.

15 + 14 = 29

Combining the constants is straightforward. 15 plus 14 equals 29.

The Simplified Expression

After combining like terms, we have $3x + 29$. And guess what? That's it! We’ve simplified the original expression $-(3x-15)+2(3x+7)$ down to its simplest form: $3x + 29$.

To recap, we used the distributive property to remove the parentheses and then combined like terms to arrive at our final simplified expression. Remember, simplifying expressions is like putting together a puzzle – each step brings you closer to the solution.

Step-by-Step Solution

Let’s quickly walk through the entire process again, just to make sure we’ve got it nailed down:

1. Distribute the -1: $-(3x-15)$ becomes $-3x + 15$.
2. Distribute the 2: $2(3x+7)$ becomes $6x + 14$.
3. Rewrite the expression: Now we have $-3x + 15 + 6x + 14$.
4. Combine like terms (x terms): $-3x + 6x = 3x$.
5. Combine like terms (constants): $15 + 14 = 29$.
6. Write the simplified expression: $3x + 29$.

See how each step logically follows the previous one? That’s the beauty of algebra! Each operation has a purpose, and when you follow the rules, you'll always find the right answer.

Common Mistakes to Avoid

Now, let’s talk about some common pitfalls that students often encounter when simplifying expressions. Being aware of these mistakes can help you avoid them and boost your confidence in algebra.

Forgetting to Distribute the Negative Sign

One of the most common errors is forgetting to distribute the negative sign properly. Remember, when you have a negative sign in front of parentheses, it’s like multiplying each term inside by -1. For example, in the expression $-(3x-15)$, it's crucial to distribute the -1 to both terms. If you only change the sign of the first term (3x) and forget to change the sign of -15, you’ll end up with an incorrect result.

So, always double-check that you’ve distributed the negative sign to every term inside the parentheses. Make it a habit to write out each step, like we did earlier, to ensure you don’t miss anything.

Incorrectly Combining Like Terms

Another common mistake is combining terms that aren’t actually “like” terms. Remember, like terms have the same variable raised to the same power. For instance, $3x$ and $6x$ are like terms because they both have ‘x’ raised to the power of 1. However, $3x$ and $3x^2$ are not like terms because the exponents are different.

When combining like terms, only add or subtract the coefficients (the numbers in front of the variables). The variable part stays the same. So, $3x + 6x$ becomes $9x$, not $9x^2$ or anything else.

Order of Operations Mix-Ups

Sometimes, students mix up the order of operations, which can lead to incorrect simplifications. Remember the acronym PEMDAS (or BODMAS) to keep the order straight:

• Parentheses (or Brackets)
• Exponents (or Orders)
• Multiplication and Division
• Addition and Subtraction

In our case, we dealt with parentheses first by distributing. Then, we combined like terms, which involved addition and subtraction. Following the correct order is crucial for accurate simplification.

Sign Errors

Sign errors are sneaky little culprits that can trip you up if you're not careful. Pay close attention to the signs (positive or negative) of each term, especially when distributing and combining. A small mistake, like writing -14 instead of +14, can throw off the entire solution.

Rushing Through the Process

Finally, one of the biggest mistakes students make is simply rushing through the process. Algebra requires patience and attention to detail. It’s better to take your time, write out each step clearly, and double-check your work than to rush and make careless errors.

Practice Problems

Okay, guys, now it's your turn to shine! Let’s put what we've learned into practice with a few practice problems. Working through these will help solidify your understanding and build your confidence.

Problem 1

Simplify: $4(2x - 3) - (x + 5)$

Problem 2

Simplify: $-2(5 - 3x) + 6x - 10$

Problem 3

Simplify: $3(x + 2) - 2(x - 1)$

Work through these problems using the steps we’ve discussed. Remember to distribute, combine like terms, and watch out for those sneaky negative signs! The answers are below, but try to solve them on your own first.

Solutions to Practice Problems

Alright, let’s see how you did! Here are the solutions to the practice problems:

Solution 1

$4(2x - 3) - (x + 5)$

• Distribute: $8x - 12 - x - 5$
• Combine like terms: $7x - 17$

Solution 2

$-2(5 - 3x) + 6x - 10$

• Distribute: $-10 + 6x + 6x - 10$
• Combine like terms: $12x - 20$

Solution 3

$3(x + 2) - 2(x - 1)$

• Distribute: $3x + 6 - 2x + 2$
• Combine like terms: $x + 8$

How did you do? If you got them all right, awesome! If you made a few mistakes, don’t worry. That’s how we learn. Go back and review the steps, see where you might have gone wrong, and try again. Practice makes perfect, guys!

Real-World Applications

You might be wondering, “Where will I ever use this in real life?” Well, simplifying expressions isn’t just an abstract math concept. It’s a fundamental skill that has applications in various fields and everyday situations. Let’s explore some real-world scenarios where simplifying expressions can come in handy.

Financial Planning

Imagine you're planning your budget. You might have a fixed income and various expenses. Simplifying expressions can help you calculate your net income (income minus expenses). For example, if your income is represented by $500 + 20x$ (where x is the number of hours you work) and your expenses are $200 + 10x$, you can simplify the expression $(500 + 20x) - (200 + 10x)$ to determine how much money you have left after expenses.

Calculating Discounts and Sales

Shopping is another area where simplifying expressions can be useful. Suppose an item is on sale for 20% off, and you have an additional coupon for $10 off. If the original price of the item is 'p', you can write an expression to represent the final price: $0.80p - 10$. Simplifying this expression (or variations of it) helps you figure out the actual cost after all discounts are applied.

Cooking and Baking

Even in the kitchen, math skills are essential! If you’re doubling or halving a recipe, you’ll need to adjust the quantities of each ingredient. Simplifying expressions can help you do this accurately. For example, if a recipe calls for $(1/2)x + 1$ cups of flour, and you want to double the recipe, you’ll need to simplify $2((1/2)x + 1)$ to find the new amount of flour needed.

Construction and Home Improvement

Construction and home improvement projects often involve measurements and calculations. Simplifying expressions can help you determine the amount of materials needed, the dimensions of a room, or the cost of a project. For example, if you’re building a rectangular fence and the length is $2x + 5$ feet and the width is $x - 3$ feet, you can simplify an expression to find the perimeter or area of the fence.

Computer Programming

In computer programming, simplifying expressions is a fundamental skill. Programmers use algebraic expressions to perform calculations, manipulate data, and control the flow of a program. Simplifying these expressions can make the code more efficient and easier to understand.

Conclusion

So, there you have it, guys! We’ve explored how to simplify the expression $-(3x-15)+2(3x+7)$, and hopefully, you’ve gained a solid understanding of the process. Remember, it’s all about breaking down the problem into manageable steps: distribute, combine like terms, and double-check your work. And now you also know where you can apply these skills in the real world!

Keep practicing, stay curious, and don’t be afraid to tackle those algebraic challenges. You’ve got this!