Simplify To Create An Equivalent Expression:$-y-3(-3y+5$\]Choose One Answer:A. $-8y-15$B. $8y+5$C. $-8y-5$D. $8y-15$

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Understanding the Problem

In this problem, we are given an expression −y−3(−3y+5)-y-3(-3y+5) and we need to simplify it to create an equivalent expression. This involves using the distributive property and combining like terms to simplify the expression.

Step 1: Apply the Distributive Property

The distributive property states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. We can use this property to simplify the expression −y−3(−3y+5)-y-3(-3y+5).

-y-3(-3y+5) = -y-(-9y-15)

Step 2: Simplify the Expression

Now, we can simplify the expression by combining like terms. We have two terms with the variable yy, so we can combine them.

-y-(-9y-15) = -y+9y+(-15)

Step 3: Combine Like Terms

We can now combine the like terms in the expression.

-y+9y+(-15) = 8y+(-15)

Step 4: Simplify the Constant Term

The constant term in the expression is −15-15. We can simplify this term by writing it as a positive number with a negative sign.

8y+(-15) = 8y-15

Conclusion

In this problem, we simplified the expression −y−3(−3y+5)-y-3(-3y+5) to create an equivalent expression. We used the distributive property and combined like terms to simplify the expression. The final simplified expression is 8y−158y-15.

Answer

The correct answer is:

  • D. 8y−158y-15

Why is this the Correct Answer?

This is the correct answer because we simplified the expression −y−3(−3y+5)-y-3(-3y+5) to create an equivalent expression, and the final simplified expression is 8y−158y-15. This is the only option that matches the simplified expression.

Tips and Tricks

  • When simplifying expressions, always use the distributive property to expand the terms.
  • Combine like terms to simplify the expression.
  • Simplify the constant term by writing it as a positive number with a negative sign.

Common Mistakes

  • Failing to use the distributive property to expand the terms.
  • Not combining like terms to simplify the expression.
  • Not simplifying the constant term.

Real-World Applications

Simplifying expressions is an important skill in mathematics and has many real-world applications. For example, in physics, we use expressions to describe the motion of objects. In economics, we use expressions to model the behavior of markets. In computer science, we use expressions to write algorithms and programs.

Conclusion

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. This means that we can distribute a single term to multiple terms inside the parentheses.

Q: How do I apply the distributive property to simplify an expression?

A: To apply the distributive property, we need to multiply the single term by each of the terms inside the parentheses. For example, if we have the expression −y−3(−3y+5)-y-3(-3y+5), we can apply the distributive property by multiplying −3-3 by each of the terms inside the parentheses: −3(−3y)-3(-3y) and −3(5)-3(5).

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property and the commutative property are two different mathematical concepts. The distributive property states that we can distribute a single term to multiple terms inside the parentheses, while the commutative property states that the order of the terms does not change the result. For example, a(b+c)=b(a+c)a(b+c) = b(a+c).

Q: How do I combine like terms to simplify an expression?

A: To combine like terms, we need to identify the terms that have the same variable and coefficient. For example, if we have the expression −y+9y+(−15)-y+9y+(-15), we can combine the like terms −y-y and 9y9y to get 8y8y.

Q: What is the difference between a like term and a unlike term?

A: A like term is a term that has the same variable and coefficient, while a unlike term is a term that has a different variable or coefficient. For example, −y-y and 9y9y are like terms, while −y-y and 55 are unlike terms.

Q: How do I simplify a constant term?

A: To simplify a constant term, we can write it as a positive number with a negative sign. For example, −15-15 can be written as −1(15)-1(15).

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is an important skill in mathematics because it helps us to:

  • Solve equations and inequalities
  • Graph functions
  • Model real-world problems
  • Write algorithms and programs

Q: How do I know if an expression is simplified?

A: An expression is simplified if it cannot be simplified further by combining like terms or applying the distributive property.

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

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

  • Failing to use the distributive property to expand the terms
  • Not combining like terms to simplify the expression
  • Not simplifying the constant term

Q: How do I apply the distributive property to simplify an expression with multiple variables?

A: To apply the distributive property to simplify an expression with multiple variables, we need to multiply each of the variables by each of the terms inside the parentheses. For example, if we have the expression −xy−3(−3xy+5)-xy-3(-3xy+5), we can apply the distributive property by multiplying −3-3 by each of the terms inside the parentheses: −3(−3xy)-3(-3xy) and −3(5)-3(5).

Q: Can I simplify an expression with a negative exponent?

A: Yes, we can simplify an expression with a negative exponent by applying the rule a−n=1ana^{-n} = \frac{1}{a^n}. For example, if we have the expression −y−2-y^{-2}, we can simplify it by applying the rule: −y−2=−1y2-y^{-2} = -\frac{1}{y^2}.

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, we need to multiply the numerator and denominator by the same value to eliminate the fraction. For example, if we have the expression −y−3\frac{-y}{-3}, we can simplify it by multiplying the numerator and denominator by −3-3: −y−3=−3y−9=y3\frac{-y}{-3} = \frac{-3y}{-9} = \frac{y}{3}.

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, we can simplify an expression with a variable in the denominator by applying the rule 1a=a−1\frac{1}{a} = a^{-1}. For example, if we have the expression −y−3\frac{-y}{-3}, we can simplify it by applying the rule: $\frac{-y}{-3} = -\frac{y}{3} = -y\frac{1}{3} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y(-3)^{-1} = -y