Select The Expressions That Are Equivalent To \[$-7c - 7d\$\].A. \[$-8d + (-4d) + (-2d)\$\] B. \[$d - 14\$\] C. \[$-8d + D - 7d\$\] D. \[$-15d + 2d\$\]

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Understanding Equivalent Expressions

In algebra, equivalent expressions are those that have the same value, even if they are written differently. To determine if two expressions are equivalent, we need to simplify them and compare their values. In this article, we will explore how to simplify algebraic expressions and select the equivalent expressions from a given set of options.

Simplifying Algebraic Expressions

To simplify an algebraic expression, we need to combine like terms and eliminate any unnecessary operations. Like terms are terms that have the same variable raised to the same power. For example, 2x and 3x are like terms because they both have the variable x raised to the power of 1.

Step 1: Combine Like Terms

To combine like terms, we need to add or subtract the coefficients of the like terms. The coefficient is the number that is multiplied by the variable. For example, in the expression 2x + 3x, the coefficients are 2 and 3. We can combine these terms by adding their coefficients: 2x + 3x = 5x.

Step 2: Eliminate Unnecessary Operations

To eliminate unnecessary operations, we need to simplify any expressions that are inside parentheses. We can do this by multiplying the expression inside the parentheses by the coefficient outside the parentheses. For example, in the expression 2(x + 3), we can eliminate the parentheses by multiplying the expression inside the parentheses by 2: 2(x + 3) = 2x + 6.

Simplifying the Given Expression

The given expression is −7c−7d{-7c - 7d}. To simplify this expression, we need to combine like terms and eliminate any unnecessary operations. However, in this case, the expression is already simplified, so we can move on to the next step.

Selecting Equivalent Expressions

Now that we have simplified the given expression, we can select the equivalent expressions from the given set of options. To do this, we need to simplify each option and compare its value to the simplified given expression.

Option A: −8d+(−4d)+(−2d){-8d + (-4d) + (-2d)}

To simplify this option, we need to combine like terms. The like terms in this option are the terms with the variable d. We can combine these terms by adding their coefficients: -8d + (-4d) + (-2d) = -14d.

Option B: d−14{d - 14}

To simplify this option, we need to eliminate any unnecessary operations. However, in this case, the option is already simplified, so we can move on to the next step.

Option C: −8d+d−7d{-8d + d - 7d}

To simplify this option, we need to combine like terms. The like terms in this option are the terms with the variable d. We can combine these terms by adding their coefficients: -8d + d - 7d = -14d.

Option D: −15d+2d{-15d + 2d}

To simplify this option, we need to combine like terms. The like terms in this option are the terms with the variable d. We can combine these terms by adding their coefficients: -15d + 2d = -13d.

Conclusion

In conclusion, the equivalent expressions to −7c−7d{-7c - 7d} are options A and C, which both simplify to -14d. Options B and D are not equivalent expressions because they simplify to -13d and -15d + 2d, respectively.

Final Answer

The final answer is:

  • Option A: −8d+(−4d)+(−2d){-8d + (-4d) + (-2d)}
  • Option C: −8d+d−7d{-8d + d - 7d}

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable raised to the same power. Unlike terms are terms that have different variables or variables raised to different powers.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. The coefficient is the number that is multiplied by the variable. For example, in the expression 2x + 3x, the coefficients are 2 and 3. You can combine these terms by adding their coefficients: 2x + 3x = 5x.

Q: What is the order of operations when simplifying algebraic expressions?

A: The order of operations when simplifying algebraic expressions is:

  1. Parentheses: Evaluate any expressions inside parentheses first.
  2. Exponents: Evaluate any exponents 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 eliminate unnecessary operations?

A: To eliminate unnecessary operations, you need to simplify any expressions that are inside parentheses. You can do this by multiplying the expression inside the parentheses by the coefficient outside the parentheses. For example, in the expression 2(x + 3), you can eliminate the parentheses by multiplying the expression inside the parentheses by 2: 2(x + 3) = 2x + 6.

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

A: A coefficient is a number that is multiplied by a variable. A constant is a number that is not multiplied by a variable.

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

A: To simplify an algebraic expression with multiple variables, you need to combine like terms and eliminate any unnecessary operations. You can do this by following the order of operations and simplifying the expression step by step.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is:

  • Option A: −8d+(−4d)+(−2d){-8d + (-4d) + (-2d)}
  • Option C: −8d+d−7d{-8d + d - 7d}

These two options are equivalent to the given expression −7c−7d{-7c - 7d}.

Common Mistakes to Avoid

  • Not combining like terms correctly
  • Not eliminating unnecessary operations
  • Not following the order of operations
  • Not simplifying expressions with multiple variables correctly

Tips for Simplifying Algebraic Expressions

  • Start by simplifying expressions inside parentheses
  • Combine like terms next
  • Eliminate unnecessary operations
  • Follow the order of operations
  • Simplify expressions with multiple variables step by step

Conclusion

In conclusion, simplifying algebraic expressions requires a step-by-step approach and attention to detail. By following the order of operations and combining like terms, you can simplify even the most complex expressions. Remember to eliminate unnecessary operations and simplify expressions with multiple variables correctly. With practice and patience, you will become proficient in simplifying algebraic expressions.