Select The Expressions That Are Equivalent To $-2(-3z-4)+9z$.A. $8z+15$B. $15z+8$C. $-4(-3z-2)+9z$D. $-4(-2z-3)+9z$

by ADMIN 116 views

=====================================================

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying a specific type of algebraic expression, namely the one given by −2(−3z−4)+9z-2(-3z-4)+9z. Our goal is to identify the equivalent expressions among the given options.

Understanding the Expression

Before we dive into simplifying the expression, let's break it down and understand its components. The given expression is −2(−3z−4)+9z-2(-3z-4)+9z. This expression consists of two main parts:

  1. Distributive Property: The expression −2(−3z−4)-2(-3z-4) involves the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac.
  2. Combining Like Terms: The expression −2(−3z−4)-2(-3z-4) can be simplified by combining like terms, which involves adding or subtracting terms with the same variable.

Simplifying the Expression

To simplify the expression −2(−3z−4)+9z-2(-3z-4)+9z, we will follow the order of operations (PEMDAS):

  1. Distribute the Negative 2: Apply the distributive property to the expression −2(−3z−4)-2(-3z-4).
  2. Combine Like Terms: Combine the like terms in the expression −2(−3z−4)-2(-3z-4).
  3. Add the Remaining Terms: Add the remaining terms, 9z9z, to the simplified expression.

Distribute the Negative 2

Using the distributive property, we can rewrite the expression −2(−3z−4)-2(-3z-4) as:

−2(−3z−4)=−2(−3z)−2(−4)-2(-3z-4) = -2(-3z) - 2(-4)

Applying the distributive property again, we get:

−2(−3z)−2(−4)=6z+8-2(-3z) - 2(-4) = 6z + 8

Combine Like Terms

Now, let's combine the like terms in the expression 6z+86z + 8:

6z+86z + 8

There are no like terms to combine, so the expression remains the same.

Add the Remaining Terms

Finally, let's add the remaining terms, 9z9z, to the simplified expression:

6z+8+9z6z + 8 + 9z

Combining the like terms, we get:

15z+815z + 8

Comparing the Simplified Expression with the Options

Now that we have simplified the expression −2(−3z−4)+9z-2(-3z-4)+9z, let's compare it with the given options:

  • Option A: 8z+158z+15
  • Option B: 15z+815z+8
  • Option C: −4(−3z−2)+9z-4(-3z-2)+9z
  • Option D: −4(−2z−3)+9z-4(-2z-3)+9z

Conclusion

Based on our simplification, we can see that the expression −2(−3z−4)+9z-2(-3z-4)+9z is equivalent to 15z+815z+8. Therefore, the correct answer is:

  • Option B: 15z+815z+8

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics, and it requires a deep understanding of the distributive property and combining like terms. By following the order of operations and applying these concepts, we can simplify complex expressions and identify equivalent expressions among the given options.

Frequently Asked Questions

  • Q: What is the distributive property? A: The distributive property is a mathematical concept that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac.
  • Q: How do I combine like terms? A: To combine like terms, add or subtract terms with the same variable.
  • Q: What is the order of operations? A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Additional Resources

  • Algebraic Expressions: A comprehensive guide to algebraic expressions, including simplifying and evaluating expressions.
  • Distributive Property: A detailed explanation of the distributive property, including examples and practice problems.
  • Combining Like Terms: A step-by-step guide to combining like terms, including examples and practice problems.

=====================================

Introduction

Algebraic expressions are a fundamental concept in mathematics, and understanding them is crucial for success in various fields. In this article, we will provide a comprehensive Q&A guide to algebraic expressions, covering topics such as simplifying and evaluating expressions, the distributive property, and combining like terms.

Q&A

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow the order of operations (PEMDAS):

  1. Distribute the coefficients: Apply the distributive property to the expression.
  2. Combine like terms: Combine the like terms in the expression.
  3. Add the remaining terms: Add the remaining terms to the simplified expression.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:

  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 combine like terms?

A: To combine like terms, add or subtract terms with the same variable.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that remains the same.

Q: How do I evaluate an algebraic expression?

A: To evaluate an algebraic expression, substitute the given values for the variables and perform the operations.

Q: What is the purpose of algebraic expressions?

A: Algebraic expressions are used to represent mathematical relationships and solve equations.

Q: How do I simplify a complex algebraic expression?

A: To simplify a complex algebraic expression, break it down into smaller parts, simplify each part, and then combine the simplified parts.

Q: What are some common algebraic expressions?

A: Some common algebraic expressions include:

  • Linear expressions: Expressions of the form ax+bax + b, where aa and bb are constants.
  • Quadratic expressions: Expressions of the form ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants.
  • Polynomial expressions: Expressions of the form anxn+an−1xn−1+…+a1x+a0a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0, where ana_n, an−1a_{n-1}, …\ldots, a1a_1, and a0a_0 are constants.

Q: How do I use algebraic expressions in real-life situations?

A: Algebraic expressions are used in various real-life situations, such as:

  • Science: Algebraic expressions are used to model and solve problems in physics, chemistry, and biology.
  • Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Algebraic expressions are used to model and analyze economic systems, such as supply and demand curves.

Conclusion

Algebraic expressions are a fundamental concept in mathematics, and understanding them is crucial for success in various fields. By following the order of operations, applying the distributive property, and combining like terms, we can simplify complex algebraic expressions and solve equations. This Q&A guide provides a comprehensive overview of algebraic expressions, covering topics such as simplifying and evaluating expressions, the distributive property, and combining like terms.

Frequently Asked Questions

  • Q: What is the difference between a variable and a constant? A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that remains the same.
  • Q: How do I evaluate an algebraic expression? A: To evaluate an algebraic expression, substitute the given values for the variables and perform the operations.
  • Q: What is the purpose of algebraic expressions? A: Algebraic expressions are used to represent mathematical relationships and solve equations.

Additional Resources

  • Algebraic Expressions: A comprehensive guide to algebraic expressions, including simplifying and evaluating expressions.
  • Distributive Property: A detailed explanation of the distributive property, including examples and practice problems.
  • Combining Like Terms: A step-by-step guide to combining like terms, including examples and practice problems.