To determine which expressions are equivalent to [tex]\(-5(-6g + 1)\)[/tex], we need to simplify each one and compare them to the initial expression. Let's go through each one step by step:
1. Simplify [tex]\(-5(-6g + 1)\)[/tex]:
[tex]\[
-5(-6g + 1) = (-5) \cdot (-6g) + (-5) \cdot 1 = 30g - 5
\][/tex]
Now let's look at each given expression to see if we can simplify them to [tex]\(30g - 5\)[/tex]:
2. Simplify [tex]\(-5(-8g + 2g + 1)\)[/tex]:
[tex]\[
-5(-8g + 2g + 1) = -5 \left(-6g + 1\right) \quad \text{(because } -8g + 2g = -6g\text{)} \\
= 30g - 5 \quad \text{(as we simplified in the first step)}
\][/tex]
Thus, [tex]\(-5(-8g + 2g + 1)\)[/tex] is equivalent to [tex]\(30g - 5\)[/tex].
3. Simplify [tex]\((-6g + 1) \cdot -5\)[/tex]:
[tex]\[
(-6g + 1) \cdot -5 = -5 \cdot (-6g) + (-5) \cdot 1 = 30g - 5
\][/tex]
Thus, [tex]\((-6g + 1) \cdot -5\)[/tex] is also equivalent to [tex]\(30g - 5\)[/tex].
4. Simplify [tex]\(-5g + 30\)[/tex]:
[tex]\[
-5g + 30 \quad \text{(which is in its simplest form)}
\][/tex]
Clearly, [tex]\(-5g + 30\)[/tex] is not equivalent to [tex]\(30g - 5\)[/tex].
5. Simplify [tex]\(30g - 5\)[/tex]:
[tex]\[
30g - 5 \quad \text{(which is already simplified)}
\][/tex]
Thus, [tex]\(30g - 5\)[/tex] is equivalent to [tex]\(30g - 5\)[/tex].
Conclusion:
The expressions that are equivalent to [tex]\(-5(-6g + 1)\)[/tex] (which simplifies to [tex]\(30g - 5\)[/tex]) are:
- [tex]\(-5(-8g + 2g + 1)\)[/tex]
- [tex]\((-6g + 1) \cdot -5\)[/tex]
- [tex]\(30g - 5\)[/tex]
