Distribute To Write An Equivalent Expression: { -\frac{1}{4}(-8x + 12y)$}$

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Understanding the Concept of Distributing

In mathematics, distributing is a fundamental concept used to simplify complex expressions. It involves multiplying each term inside the parentheses by the factor outside the parentheses. In this article, we will focus on distributing to write an equivalent expression for the given problem: βˆ’14(βˆ’8x+12y){-\frac{1}{4}(-8x + 12y)}

The Distributive Property

The distributive property is a mathematical concept that states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property allows us to distribute a single term to multiple terms inside the parentheses.

Applying the Distributive Property

To write an equivalent expression for the given problem, we will apply the distributive property. We will multiply each term inside the parentheses by the factor outside the parentheses, which is βˆ’14{-\frac{1}{4}}

Step 1: Multiply the first term

The first term inside the parentheses is βˆ’8x{-8x}. We will multiply this term by the factor βˆ’14{-\frac{1}{4}}

βˆ’14(βˆ’8x)=14(8x)=2x{-\frac{1}{4}(-8x) = \frac{1}{4}(8x) = 2x}

Step 2: Multiply the second term

The second term inside the parentheses is 12y{12y}. We will multiply this term by the factor βˆ’14{-\frac{1}{4}}

βˆ’14(12y)=14(12y)=3y{-\frac{1}{4}(12y) = \frac{1}{4}(12y) = 3y}

Step 3: Combine the terms

Now that we have multiplied each term inside the parentheses by the factor outside the parentheses, we can combine the terms to write an equivalent expression.

2x+3y{2x + 3y}

Conclusion

In this article, we applied the distributive property to write an equivalent expression for the given problem. We multiplied each term inside the parentheses by the factor outside the parentheses and combined the terms to simplify the expression. The final equivalent expression is 2x+3y{2x + 3y}

Example Problems

Problem 1

Write an equivalent expression for the following problem: 12(4xβˆ’6y){\frac{1}{2}(4x - 6y)}

Solution

To write an equivalent expression, we will apply the distributive property. We will multiply each term inside the parentheses by the factor outside the parentheses, which is 12{\frac{1}{2}}

12(4x)=2x{\frac{1}{2}(4x) = 2x}

12(βˆ’6y)=βˆ’3y{\frac{1}{2}(-6y) = -3y}

Combining the terms, we get:

2xβˆ’3y{2x - 3y}

Problem 2

Write an equivalent expression for the following problem: βˆ’3(2x+5y){-3(2x + 5y)}

Solution

To write an equivalent expression, we will apply the distributive property. We will multiply each term inside the parentheses by the factor outside the parentheses, which is βˆ’3{-3}

βˆ’3(2x)=βˆ’6x{-3(2x) = -6x}

βˆ’3(5y)=βˆ’15y{-3(5y) = -15y}

Combining the terms, we get:

βˆ’6xβˆ’15y{-6x - 15y}

Tips and Tricks

  • When applying the distributive property, make sure to multiply each term inside the parentheses by the factor outside the parentheses.
  • Use the distributive property to simplify complex expressions and make them easier to work with.
  • Practice applying the distributive property to different types of expressions to become more comfortable with the concept.

Common Mistakes

  • Failing to multiply each term inside the parentheses by the factor outside the parentheses.
  • Not combining the terms after multiplying each term by the factor.
  • Not using the distributive property to simplify complex expressions.

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions about distributing to write an equivalent expression.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property allows us to distribute a single term to multiple terms inside the parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses by the factor outside the parentheses. For example, if you have the expression βˆ’14(βˆ’8x+12y){-\frac{1}{4}(-8x + 12y)}, you would multiply each term inside the parentheses by βˆ’14{-\frac{1}{4}}

Q: What if I have a negative sign outside the parentheses?

A: If you have a negative sign outside the parentheses, you need to multiply each term inside the parentheses by the negative sign. For example, if you have the expression βˆ’3(2x+5y){-3(2x + 5y)}, you would multiply each term inside the parentheses by βˆ’3{-3}

Q: Can I use the distributive property with fractions?

A: Yes, you can use the distributive property with fractions. For example, if you have the expression 12(4xβˆ’6y){\frac{1}{2}(4x - 6y)}, you would multiply each term inside the parentheses by 12{\frac{1}{2}}

Q: What if I have a variable inside the parentheses?

A: If you have a variable inside the parentheses, you need to multiply the variable by the factor outside the parentheses. For example, if you have the expression βˆ’14(2x+3y){-\frac{1}{4}(2x + 3y)}, you would multiply the variable x{x} by βˆ’14{-\frac{1}{4}} and the variable y{y} by βˆ’14{-\frac{1}{4}}

Q: Can I use the distributive property with decimals?

A: Yes, you can use the distributive property with decimals. For example, if you have the expression βˆ’0.5(4xβˆ’6y){-0.5(4x - 6y)}, you would multiply each term inside the parentheses by βˆ’0.5{-0.5}

Q: What if I have a mixed expression?

A: If you have a mixed expression, you need to apply the distributive property to each term separately. For example, if you have the expression βˆ’14(βˆ’8x+12y+3z){-\frac{1}{4}(-8x + 12y + 3z)}, you would multiply each term inside the parentheses by βˆ’14{-\frac{1}{4}}

Q: Can I use the distributive property with exponents?

A: Yes, you can use the distributive property with exponents. For example, if you have the expression βˆ’3(2x2+5y2){-3(2x^2 + 5y^2)}, you would multiply each term inside the parentheses by βˆ’3{-3}

Conclusion

In this article, we answered some frequently asked questions about distributing to write an equivalent expression. We covered topics such as the distributive property, applying the distributive property, and using the distributive property with fractions, variables, decimals, and exponents.

Tips and Tricks

  • Make sure to multiply each term inside the parentheses by the factor outside the parentheses.
  • Use the distributive property to simplify complex expressions and make them easier to work with.
  • Practice applying the distributive property to different types of expressions to become more comfortable with the concept.

Common Mistakes

  • Failing to multiply each term inside the parentheses by the factor outside the parentheses.
  • Not combining the terms after multiplying each term by the factor.
  • Not using the distributive property to simplify complex expressions.

Example Problems

Problem 1

Write an equivalent expression for the following problem: 13(6xβˆ’9y){\frac{1}{3}(6x - 9y)}

Solution

To write an equivalent expression, we will apply the distributive property. We will multiply each term inside the parentheses by the factor outside the parentheses, which is 13{\frac{1}{3}}

13(6x)=2x{\frac{1}{3}(6x) = 2x}

13(βˆ’9y)=βˆ’3y{\frac{1}{3}(-9y) = -3y}

Combining the terms, we get:

2xβˆ’3y{2x - 3y}

Problem 2

Write an equivalent expression for the following problem: βˆ’2(3x+4y){-2(3x + 4y)}

Solution

To write an equivalent expression, we will apply the distributive property. We will multiply each term inside the parentheses by the factor outside the parentheses, which is βˆ’2{-2}

βˆ’2(3x)=βˆ’6x{-2(3x) = -6x}

βˆ’2(4y)=βˆ’8y{-2(4y) = -8y}

Combining the terms, we get:

βˆ’6xβˆ’8y{-6x - 8y}

Conclusion

In this article, we answered some frequently asked questions about distributing to write an equivalent expression. We covered topics such as the distributive property, applying the distributive property, and using the distributive property with fractions, variables, decimals, and exponents.