To determine which expressions are equivalent to [tex]\(35 + 30s - 45t\)[/tex], we need to simplify each given expression and compare them to [tex]\(35 + 30s - 45t\)[/tex].
### Expression A: [tex]\(7 \cdot (5 + 30s - 45t)\)[/tex]
Let's expand this expression:
[tex]\[ 7 \cdot (5 + 30s - 45t) = 7 \cdot 5 + 7 \cdot 30s - 7 \cdot 45t \][/tex]
[tex]\[ = 35 + 210s - 315t \][/tex]
Since [tex]\(35 + 210s - 315t\)[/tex] is not equivalent to [tex]\(35 + 30s - 45t\)[/tex], Expression A is not equivalent.
### Expression B: [tex]\(5(7 + 6s - 9t)\)[/tex]
Similarly, expand this expression:
[tex]\[ 5(7 + 6s - 9t) = 5 \cdot 7 + 5 \cdot 6s - 5 \cdot 9t \][/tex]
[tex]\[ = 35 + 30s - 45t \][/tex]
This matches [tex]\(35 + 30s - 45t\)[/tex] exactly. Therefore, Expression B is equivalent.
### Expression C: [tex]\((-35 - 30s + 45t) \times (-1)\)[/tex]
Let's multiply:
[tex]\[ (-35 - 30s + 45t) \times (-1) = -1 \cdot (-35) + (-1) \cdot (-30s) + (-1) \cdot (45t) \][/tex]
[tex]\[ = 35 + 30s - 45t \][/tex]
This matches [tex]\(35 + 30s - 45t\)[/tex] exactly. Therefore, Expression C is equivalent.
### Expression D: [tex]\(10 \times (3.5 + 3s - 4.5)\)[/tex]
Expand this expression:
[tex]\[ 10 \times (3.5 + 3s - 4.5) = 10 \cdot 3.5 + 10 \cdot 3s - 10 \cdot 4.5 \][/tex]
[tex]\[ = 35 + 30s - 45 \][/tex]
Since [tex]\(35 + 30s - 45\)[/tex] is not equivalent to [tex]\(35 + 30s - 45t\)[/tex], Expression D is not equivalent.
### Expression E: [tex]\(\left(\frac{35}{2} - 15s + \frac{45}{2}t\right) \times (-2)\)[/tex]
Let's expand this expression:
[tex]\[ \left(\frac{35}{2} - 15s + \frac{45}{2}t\right) \times (-2) = -2 \cdot \frac{35}{2} + (-2) \cdot (-15s) + (-2) \cdot \frac{45}{2}t \][/tex]
[tex]\[ = -35 + 30s - 45t \][/tex]
Since [tex]\(-35 + 30s - 45t\)[/tex] is not equivalent to [tex]\(35 + 30s - 45t\)[/tex] (the sign on the constant term differs), Expression E is not equivalent.
### Conclusion
Based on our calculations, the two expressions that are equivalent to [tex]\(35 + 30s - 45t\)[/tex] are:
B and C.
