To determine which expressions are equivalent to [tex]\( 5(9r - 2) + 3 \)[/tex], let's simplify and compare each given expression step-by-step.
1. Given Expression:
[tex]\[
5(9r - 2) + 3
\][/tex]
Simplifying the given expression:
[tex]\[
5(9r - 2) + 3 = 5 \cdot 9r - 5 \cdot 2 + 3 = 45r - 10 + 3 = 45r - 7
\][/tex]
So the simplified form of the given expression is [tex]\( 45r - 7 \)[/tex].
2. First Expression:
[tex]\[
45r - 7
\][/tex]
This expression matches [tex]\( 45r - 7 \)[/tex], so it is equivalent to the given expression.
3. Second Expression:
[tex]\[
(9r - 2) \cdot 5 + 3
\][/tex]
Simplifying this expression:
[tex]\[
(9r - 2) \cdot 5 + 3 = 9r \cdot 5 - 2 \cdot 5 + 3 = 45r - 10 + 3 = 45r - 7
\][/tex]
This expression also simplifies to [tex]\( 45r - 7 \)[/tex], so it is equivalent to the given expression.
4. Third Expression:
[tex]\[
5(-2 + 9r) + 3
\][/tex]
Simplifying this expression:
[tex]\[
5(-2 + 9r) + 3 = 5 \cdot (-2) + 5 \cdot 9r + 3 = -10 + 45r + 3 = 45r - 10 + 3 = 45r - 7
\][/tex]
This expression again simplifies to [tex]\( 45r - 7 \)[/tex], so it is equivalent to the given expression.
5. Fourth Expression:
[tex]\[
9(-2r + 5) + 3
\][/tex]
Simplifying this expression:
[tex]\[
9(-2r + 5) + 3 = 9 \cdot (-2r) + 9 \cdot 5 + 3 = -18r + 45 + 3 = -18r + 48
\][/tex]
This does not simplify to [tex]\( 45r - 7 \)[/tex], so it is not equivalent to the given expression.
Conclusion:
The expressions that are equivalent to [tex]\( 5(9r - 2) + 3 \)[/tex] are:
- [tex]\( 45r - 7 \)[/tex]
- [tex]\( (9r - 2) \cdot 5 + 3 \)[/tex]
- [tex]\( 5(-2 + 9r) + 3 \)[/tex]
The expression [tex]\( 9(-2r + 5) + 3 \)[/tex] is not equivalent to the given expression.
