Expand And Simplify $3(2x+1)+2(x+4)$.

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given algebraic expression $3(2x+1)+2(x+4)$. We will break down the process into manageable steps, making it easy to understand and follow.

Understanding the Expression

Before we start simplifying the expression, let's take a closer look at it. The given expression is $3(2x+1)+2(x+4)$. This expression consists of two terms, each containing a product of a constant and a variable. The first term is $3(2x+1)$, and the second term is $2(x+4)$.

Distributive Property

To simplify the expression, we will use the distributive property, which states that for any real numbers a, b, and c, the following equation holds:

a(b+c)=ab+aca(b+c) = ab + ac

Using this property, we can expand the first term $3(2x+1)$ as follows:

3(2x+1)=3â‹…2x+3â‹…13(2x+1) = 3 \cdot 2x + 3 \cdot 1

=6x+3= 6x + 3

Similarly, we can expand the second term $2(x+4)$ as follows:

2(x+4)=2â‹…x+2â‹…42(x+4) = 2 \cdot x + 2 \cdot 4

=2x+8= 2x + 8

Combining Like Terms

Now that we have expanded both terms, we can combine like terms to simplify the expression further. Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms: $6x$ and $2x$.

To combine these like terms, we add their coefficients:

6x+2x=(6+2)x6x + 2x = (6+2)x

=8x= 8x

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression by adding the constants:

3+8=113 + 8 = 11

So, the simplified expression is:

8x+118x + 11

Conclusion

In this article, we have simplified the given algebraic expression $3(2x+1)+2(x+4)$ using the distributive property and combining like terms. We have shown that the simplified expression is $8x + 11$. This process demonstrates the importance of simplifying algebraic expressions, as it makes it easier to work with and understand the underlying mathematics.

Tips and Tricks

  • When simplifying algebraic expressions, always look for opportunities to use the distributive property.
  • Combine like terms to simplify the expression further.
  • Use the distributive property to expand terms, and then combine like terms to simplify the expression.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. In economics, algebraic expressions are used to model economic systems. In computer science, algebraic expressions are used to write algorithms and solve problems.

Common Mistakes

When simplifying algebraic expressions, there are several common mistakes to avoid:

  • Failing to use the distributive property when expanding terms.
  • Not combining like terms to simplify the expression further.
  • Making errors when adding or subtracting constants.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

  • Simplify the expression $4(3x-2)+2(x+5)$.
  • Simplify the expression $2(5x+3)+3(x-2)$.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By using the distributive property and combining like terms, we can simplify complex expressions and make them easier to work with. Remember to always look for opportunities to use the distributive property and combine like terms to simplify the expression further. With practice and patience, you will become proficient in simplifying algebraic expressions and be able to tackle even the most complex problems.

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Q: What is the distributive property, and how is it used in simplifying algebraic expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, the following equation holds:

a(b+c)=ab+aca(b+c) = ab + ac

This property is used to expand terms in an algebraic expression by multiplying the constant with each term inside the parentheses.

Q: How do I simplify an algebraic expression using the distributive property?

A: To simplify an algebraic expression using the distributive property, follow these steps:

  1. Identify the terms inside the parentheses.
  2. Multiply the constant with each term inside the parentheses.
  3. Combine like terms to simplify the expression further.

Q: What are like terms, and how are they combined in simplifying algebraic expressions?

A: Like terms are terms that have the same variable raised to the same power. In simplifying algebraic expressions, like terms are combined by adding their coefficients.

Q: How do I identify like terms in an algebraic expression?

A: To identify like terms in an algebraic expression, look for terms that have the same variable raised to the same power. For example, in the expression $6x + 2x$, the terms $6x$ and $2x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: What is the difference between a constant and a variable in an algebraic expression?

A: In an algebraic expression, a constant is a number that does not change value, while a variable is a letter or symbol that represents a value that can change.

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, follow these steps:

  1. Identify the terms with each variable.
  2. Combine like terms for each variable.
  3. Simplify the expression further by combining like terms.

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

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

  • Failing to use the distributive property when expanding terms.
  • Not combining like terms to simplify the expression further.
  • Making errors when adding or subtracting constants.

Q: How do I check my work when simplifying an algebraic expression?

A: To check your work when simplifying an algebraic expression, follow these steps:

  1. Simplify the expression using the distributive property and combining like terms.
  2. Check that the simplified expression is equivalent to the original expression.
  3. Verify that the simplified expression is in its simplest form.

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

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

  • Physics: Algebraic expressions are used to describe the motion of objects.
  • Economics: Algebraic expressions are used to model economic systems.
  • Computer Science: Algebraic expressions are used to write algorithms and solve problems.

Q: How can I practice simplifying algebraic expressions?

A: To practice simplifying algebraic expressions, try the following:

  • Simplify algebraic expressions with different variables and constants.
  • Use online resources and practice problems to test your skills.
  • Work with a partner or tutor to review and practice simplifying algebraic expressions.