Equation For Sums, Differences: A Math Problem Solved!

Hey everyone! Let's break down this math problem step by step to figure out the correct equation. This type of problem involves translating a word problem into an algebraic equation, which is a fundamental skill in mathematics. We need to carefully consider each part of the statement to accurately represent it in mathematical symbols.

Understanding the Problem

The problem statement is: "six added to twice the sum of a number and four is equal to one-half of the difference of three and the number".

Let's dissect this sentence piece by piece:

1. "a number": This is our variable. We'll call it x. You can use any letter, but x is pretty common.
2. "the sum of a number and four": This means we add 4 to our number x, so it's (x + 4).
3. "twice the sum of a number and four": We multiply the sum (x + 4) by 2, giving us 2(x + 4).
4. "six added to twice the sum of a number and four": We add 6 to the previous expression, so we have 6 + 2(x + 4). This is the left side of our equation.
5. "the difference of three and the number": This means we subtract the number x from 3, which is (3 - x).
6. "one-half of the difference of three and the number": We multiply the difference (3 - x) by 1/2, giving us (1/2)(3 - x). This is the right side of our equation.
7. "is equal to": This translates directly to the equals sign (=).

So, putting it all together, we get the equation: 6 + 2(x + 4) = (1/2)(3 - x).

Analyzing the Options

Now, let's look at the given options and see which one matches our derived equation:

A. $6+2(x+4)=\frac{1}{2}(3-x)$ B. $6+2(x+4)=\frac{1}{2}(x-3)$ C. $(6+2)(x+4)=\frac{1}{2}(3-x)$

Option A looks exactly like what we found: 6 + 2(x + 4) = (1/2)(3 - x). So, this is likely the correct answer.

Option B has a slight difference on the right side: 6 + 2(x + 4) = (1/2)(x - 3). Notice that (x - 3) is not the same as (3 - x). The order of subtraction matters!

Option C changes the left side of the equation: (6 + 2)(x + 4) = (1/2)(3 - x). Here, 6 and 2 are added before multiplying by (x + 4), which is not what the original problem statement says. The statement says to add six to twice the sum, not to add 6 and 2 together and then multiply.

The Correct Equation

Based on our analysis, the correct equation that represents the given statement is:

A. $6+2(x+4)=\frac{1}{2}(3-x)$

This equation accurately captures all the relationships described in the word problem. It correctly represents adding six to twice the sum of a number and four, and equates it to one-half of the difference of three and the number.

Tips for Solving Similar Problems

When tackling these types of problems, remember these key tips:

• Read Carefully: Make sure you understand every word and phrase in the problem statement.
• Break It Down: Dissect the sentence into smaller, manageable parts.
• Define the Variable: Clearly identify what the variable represents.
• Translate Step by Step: Convert each part of the sentence into a mathematical expression or operation.
• Check Your Work: After forming the equation, reread the problem to ensure your equation accurately reflects the given information.

By following these steps, you can confidently translate word problems into algebraic equations and solve them effectively. Keep practicing, and you'll become a pro at these in no time! These skills are super useful in all sorts of math and science situations. Keep an eye out for keywords like "sum," "difference," "product," and "quotient," as these indicate specific mathematical operations.

Additional Practice

To further solidify your understanding, try creating your own word problems and translating them into equations. You can also find plenty of practice problems online or in textbooks. Remember, the more you practice, the better you'll become at this skill. You might encounter phrases like "increased by," which means addition, or "decreased by," which means subtraction. Understanding these common phrases is super helpful.

Also, pay attention to the order of operations. In the problem we solved, it was crucial to recognize that we needed to find the sum of x and 4 before multiplying by 2. Ignoring the order of operations can lead to an incorrect equation. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to help you keep the order straight.

Conclusion

In conclusion, translating word problems into algebraic equations is a critical skill in mathematics. By carefully breaking down the problem statement, defining the variable, and translating each part step by step, you can accurately represent the given information in an equation. Always double-check your work to ensure that your equation reflects the relationships described in the problem. With practice and attention to detail, you can master this skill and confidently solve a wide range of mathematical problems. Keep up the great work, and you'll be solving complex equations like a pro in no time! You've got this!