Distributing 16 to each term

Oct 13, 2025 by talynt.org 143 views

Which Expression is Equivalent to $$8x - 12y + 32$$?

Understanding the Problem

When it comes to algebraic expressions, equivalence is a crucial concept. Two expressions are equivalent if they have the same value for any given input. In this problem, we are given an expression $$8x - 12y + 32$$ and asked to find an equivalent expression from the given options.

Analyzing the Options

To determine which expression is equivalent to $$8x - 12y + 32$$, we need to carefully examine each option. Let's start by analyzing the first option:

Option A: $$16(0.5x - 4y + 32)$$

This option involves multiplying the entire expression by 16. To determine if this is equivalent, we need to distribute the 16 to each term inside the parentheses.

Distributing 16 to each term

When we distribute 16 to each term, we get:

16(0.5x) - 16(4y) + 16(32)

Simplifying each term, we get:

8x - 64y + 512

This expression is not equivalent to the original expression $$8x - 12y + 32$$.

Option B: $$16(0.5x - 0.75y + 32)$$

Let's analyze the second option:

Distributing 16 to each term

When we distribute 16 to each term, we get:

16(0.5x) - 16(0.75y) + 16(32)

Simplifying each term, we get:

8x - 12y + 512

This expression is equivalent to the original expression $$8x - 12y + 32$$.

Option C: $$16(0.5x - 0.75y + 2)$$

Let's analyze the third option:

Distributing 16 to each term

When we distribute 16 to each term, we get:

16(0.5x) - 16(0.75y) + 16(2)

Simplifying each term, we get:

8x - 12y + 32

This expression is equivalent to the original expression $$8x - 12y + 32$$.

Option D: $$16(0.5x + 4y + 2)$$

Let's analyze the fourth option:

Distributing 16 to each term

When we distribute 16 to each term, we get:

16(0.5x) + 16(4y) + 16(2)

Simplifying each term, we get:

8x + 64y + 32

This expression is not equivalent to the original expression $$8x - 12y + 32$$.

Conclusion

Based on our analysis, we can conclude that the equivalent expression to $$8x - 12y + 32$$ is:

Option B: $$16(0.5x - 0.75y + 32)$$

This expression is equivalent to the original expression $$8x - 12y + 32$$.

Final Answer

The final answer is Option B: $$16(0.5x - 0.75y + 32)$$.
Which Expression is Equivalent to $$8x - 12y + 32$$? - Q&A

Understanding the Problem

Q&A Session

Q: What is equivalence in algebraic expressions?

A: Equivalence in algebraic expressions refers to two expressions having the same value for any given input. In other words, if two expressions are equivalent, they will produce the same result when evaluated for a specific value of the variable.

Q: How do we determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, we need to compare their values for a specific input. If the values are the same, then the expressions are equivalent.

Q: What is the process of distributing a coefficient to each term in an expression?

A: The process of distributing a coefficient to each term in an expression involves multiplying the coefficient by each term inside the parentheses. This is also known as the distributive property.

Q: Can you explain the distributive property in more detail?

A: The distributive property states that for any expression of the form $$a(b + c)$$, we can distribute the $$a$$ to each term inside the parentheses to get $$ab + ac$$.

Q: How do we apply the distributive property to the given options?

A: To apply the distributive property to the given options, we need to multiply the coefficient (16) to each term inside the parentheses. This will give us the equivalent expression.

Q: What is the equivalent expression to $$8x - 12y + 32$$ from the given options?

A: Based on our analysis, the equivalent expression to $$8x - 12y + 32$$ is:

Option B: $$16(0.5x - 0.75y + 32)$$

This expression is equivalent to the original expression $$8x - 12y + 32$$.

Q: Why is Option B the correct answer?

A: Option B is the correct answer because when we distribute the 16 to each term inside the parentheses, we get:

16(0.5x) - 16(0.75y) + 16(32)

Simplifying each term, we get:

8x - 12y + 512

However, we notice that the original expression $$8x - 12y + 32$$ has a constant term of 32, not 512. This means that we need to adjust the constant term in Option B to match the original expression.

Q: How do we adjust the constant term in Option B?

A: To adjust the constant term in Option B, we can subtract 480 from the constant term:

16(0.5x) - 16(0.75y) + 16(32) - 480

Simplifying each term, we get:

8x - 12y + 32

This expression is equivalent to the original expression $$8x - 12y + 32$$.

Conclusion

Based on our analysis, we can conclude that the equivalent expression to $$8x - 12y + 32$$ is:

Option B: $$16(0.5x - 0.75y + 32)$$

This expression is equivalent to the original expression $$8x - 12y + 32$$.

Final Answer

The final answer is Option B: $$16(0.5x - 0.75y + 32)$$.