Which expressions are equivalent to 10p  18? Select TWO correct answers.
Understand the Problem
The question is asking which expressions are equivalent to the expression 10p  18. The goal is to select two correct answers from the given options based on algebraic manipulation.
Answer
$\frac{1}{2}(20p  36)$ and $2(5p  9)$
Answer for screen readers
The two expressions equivalent to $10p  18$ are:
 $\frac{1}{2}(20p  36)$
 $2(5p  9)$
Steps to Solve

Identify the Target Expression We are looking for expressions that are equivalent to $10p  18$.

Simplify Each Option We'll simplify each given expression to check for equivalence.

Option 1: $\frac{1}{2}(20p  36)$
 Distributing $\frac{1}{2}$: $$ \frac{1}{2} \cdot 20p  \frac{1}{2} \cdot 36 = 10p  18 $$
 This matches our target expression.

Option 2: $2(5p + 9)$
 Distributing $2$: $$ 2 \cdot 5p + 2 \cdot 9 = 10p + 18 $$
 This does not match our target expression.

Option 3: $\frac{1}{2}(5p  9)$
 Distributing $\frac{1}{2}$: $$ \frac{1}{2} \cdot 5p  \frac{1}{2} \cdot 9 = \frac{5p}{2}  \frac{9}{2} $$
 This does not match our target expression.

Option 4: $2(5p  9)$
 Distributing $2$: $$ 2 \cdot 5p  2 \cdot 9 = 10p  18 $$
 This matches our target expression.

Option 5: $2(5p)$
 Distributing $2$: $$ 2 \cdot 5p  2 \cdot 9 = 10p  18 $$
 This matches our target expression.


Conclusion of Options The two expressions equivalent to $10p  18$ are:
 Option 1: $\frac{1}{2}(20p  36)$
 Option 4: $2(5p  9)$
The two expressions equivalent to $10p  18$ are:
 $\frac{1}{2}(20p  36)$
 $2(5p  9)$
More Information
Both selected expressions simplify to the same original expression by correctly applying the distributive property. This demonstrates the concept of expression equivalence through algebraic manipulation.
Tips
 Confusing the signs when distributing, particularly with negatives (e.g., turning $9$ into $+9$).
 Failing to simplify correctly, especially if not fully applying the distribution step.