Let's verify Bobby's claim step by step by using the information provided and analyzing the equations and results carefully.
1. Substitute 21 for \( x \) in the original equation:
The original equation is:
[tex]\[
\frac{1}{2}(x - 3) = 9
\][/tex]
Substitute \( x = 21 \):
[tex]\[
\frac{1}{2}(21 - 3) = \frac{1}{2}(18) = 9
\][/tex]
This substitution holds true as:
[tex]\[
9 = 9
\][/tex]
Hence, substituting 21 for \( x \) in the original equation verifies the claim correctly.
2. Substitute any number for \( x \) in the original equation:
In general, you can substitute any number for \( x \), but it may not satisfy the equation. Here, we are specifically testing for \( x = 21 \) to verify Bobby’s claim.
3. Bobby's claim that \( x = 21 \) is correct:
Since:
[tex]\[
\frac{1}{2}(21 - 3) = 9
\][/tex]
And simplifying gives:
[tex]\[
\frac{1}{2}(18) = 9
\][/tex]
Which results in:
[tex]\[
9 = 9
\][/tex]
This indicates that Bobby’s claim is indeed correct. Therefore, Peter started with 21 cards.
4. The result of verifying Bobby's work is \( 9 = 9 \):
From the verification steps we see:
[tex]\[
\frac{1}{2}(21 - 3) = 9
\][/tex]
Results in:
[tex]\[
9 = 9
\][/tex]
Hence, \( 9 = 9 \) is a true statement validating Bobby's work.
5. The result of verifying Bobby's work is \( 21 = 21 \):
This is not applicable in the context of the original equation, as there is no step in the verification that directly results in \( 21 = 21 \). Therefore, this statement is false.
In summary:
- \( \boxed{\text{Substitute 21 for } x \text{ in the original equation.}} \)
- \( \boxed{\text{Bobby's claim that } x = 21 \text{ is correct.}} \)
- \( \boxed{\text{The result of verifying Bobby's work is } 9 = 9.} \)
The other statements do not apply in verifying Bobby's claim.
