What Is The Simplified Expression For The Expression Below?$\[ \frac{1}{2}(8x + 4) + \frac{1}{3}(9 - 3x) \\]A. \[$5x + 5\$\]B. \[$7x + 1\$\]C. \[$x + 7\$\]D. \[$3x + 5\$\]

Understanding the Problem

The given problem involves simplifying an algebraic expression, which is a fundamental concept in mathematics. The expression to be simplified is $\frac{1}{2}(8x + 4) + \frac{1}{3}(9 - 3x)$. To simplify this expression, we need to apply the rules of algebra, which include the distributive property, combining like terms, and simplifying fractions.

Step 1: Apply the Distributive Property

The first step in simplifying the given expression is to apply the distributive property. This property states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We can apply this property to the first term of the expression, $\frac{1}{2}(8x + 4)$.

\frac{1}{2}(8x + 4) = \frac{1}{2} \cdot 8x + \frac{1}{2} \cdot 4

Step 2: Simplify the First Term

Now that we have applied the distributive property to the first term, we can simplify it further. We can simplify the first term by multiplying the fraction $\frac{1}{2}$ with the terms inside the parentheses.

\frac{1}{2} \cdot 8x = 4x
\frac{1}{2} \cdot 4 = 2

Step 3: Apply the Distributive Property to the Second Term

The next step is to apply the distributive property to the second term of the expression, $\frac{1}{3}(9 - 3x)$. We can apply this property by multiplying the fraction $\frac{1}{3}$ with the terms inside the parentheses.

\frac{1}{3}(9 - 3x) = \frac{1}{3} \cdot 9 - \frac{1}{3} \cdot 3x

Step 4: Simplify the Second Term

Now that we have applied the distributive property to the second term, we can simplify it further. We can simplify the second term by multiplying the fraction $\frac{1}{3}$ with the terms inside the parentheses.

\frac{1}{3} \cdot 9 = 3
\frac{1}{3} \cdot 3x = x

Step 5: Combine Like Terms

Now that we have simplified both terms, we can combine like terms. We can combine the like terms by adding or subtracting the coefficients of the same variables.

4x + 2 + 3 - x = 3x + 5

Conclusion

In conclusion, the simplified expression for the given algebraic expression is $3x + 5$. This is the final answer to the problem.

Discussion

The given problem is a fundamental concept in mathematics, and it requires the application of the distributive property, combining like terms, and simplifying fractions. The problem is designed to test the student's understanding of algebraic expressions and their simplification.

Answer Key

The answer key to the problem is:

• A. $5x + 5$
• B. $7x + 1$
• C. $x + 7$
• D. $3x + 5$

The correct answer is D. $3x + 5$.

Final Answer

The final answer to the problem is $3x + 5$.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: What is the distributive property in algebra?

A: The distributive property in algebra is a rule that allows us to multiply a single term to multiple terms inside parentheses. It states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to apply the rules of algebra, which include the distributive property, combining like terms, and simplifying fractions.

Q: What are like terms in algebra?

A: Like terms in algebra are terms that have the same variable raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the same variables. For example, $2x + 5x = 7x$.

Q: What is the order of operations in algebra?

A: The order of operations in algebra is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify fractions in algebra?

A: To simplify fractions in algebra, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I use the GCD to simplify fractions?

A: To use the GCD to simplify fractions, you need to find the GCD of the numerator and denominator and divide both numbers by the GCD.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Forgetting to apply the distributive property
• Not combining like terms
• Not simplifying fractions
• Not following the order of operations

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through practice problems and exercises. You can also use online resources and tools to help you practice and learn.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

• Solving systems of equations
• Finding the maximum or minimum value of a function
• Modeling real-world phenomena
• Optimizing business decisions

Q: How can I use technology to help me simplify algebraic expressions?

A: You can use technology, such as calculators and computer algebra systems, to help you simplify algebraic expressions. These tools can perform calculations and simplify expressions quickly and accurately.

Q: What are some tips for simplifying algebraic expressions?

A: Some tips for simplifying algebraic expressions include:

• Start by simplifying the expression inside the parentheses
• Use the distributive property to simplify the expression
• Combine like terms
• Simplify fractions
• Follow the order of operations

Q: How can I check my work when simplifying algebraic expressions?

A: You can check your work by plugging the simplified expression back into the original equation and verifying that it is true. You can also use technology, such as calculators and computer algebra systems, to check your work.