To determine which expressions are equivalent to \(-2(7a-1)-3a\), we will simplify the original expression and then each of the options.
- Original expression: \[ -2(7a - 1) - 3a = -2 \cdot 7a + 2 \cdot 1 - 3a = -14a + 2 - 3a = -14a - 3a + 2 = -17a + 2 \]
Now let's simplify each of the options:
-
\(-2(-9a + 16a - 1) - 3a\): \[ -2(-9a + 16a - 1) = -2(7a - 1) = -2 \cdot 7a + 2 \cdot 1 = -14a + 2 \] Then subtracting \(3a\): \[ -14a + 2 - 3a = -14a - 3a + 2 = -17a + 2 \]
-
\(-2(-1 + 7a) - 3a\): \[ -2(-1 + 7a) = -2 \cdot (-1) + -2 \cdot (7a) = 2 - 14a \] Then subtracting \(3a\): \[ 2 - 14a - 3a = 2 - 17a \]
-
\(-2(3a + 4a - 1) - 3a\): \[ -2(3a + 4a - 1) = -2(7a - 1) = -14a + 2 \] Then subtracting \(3a\): \[ -14a + 2 - 3a = -14a - 3a + 2 = -17a + 2 \]
-
\(-17a + 2\): This expression is already in the form we derived earlier.
Now, let's compare the results:
- Option 1: \(-17a + 2\) (equivalent to original)
- Option 2: \(2 - 17a\) (not equivalent)
- Option 3: \(-17a + 2\) (equivalent to original)
- Option 4: \(-17a + 2\) (equivalent to original)
Thus, the equivalent expressions are: 1, 3, and 4.