To determine which expressions are equivalent to \( 4(x + 3) - 10 + 6x \), we first need to simplify this expression step by step.
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Distribute the \( 4 \) to \( (x + 3) \):
\[ 4(x + 3) = 4x + 12 \]
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Substitute back into the expression:
\[ 4x + 12 - 10 + 6x \]
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Combine like terms:
\[ 4x + 6x + 12 - 10 = 10x + 2 \]
So, \( 4(x + 3) - 10 + 6x \) simplifies to \( 10x + 2 \).
Now let's compare this with the provided options:
- \( 10x - 7 \): Not equivalent (it differs by +9).
- \( -2x - 7 \): Not equivalent (wrong direction and different coefficients).
- \( 2(5x + 1) \): Let's simplify this: \[ 2(5x + 1) = 10x + 2 \quad \text{(Equivalent)} \]
- \( 2(8x + 1) \): Let's simplify this: \[ 2(8x + 1) = 16x + 2 \quad \text{(Not equivalent)} \]
- \( 4x + 3 - 10 + 6x \): Let's simplify: \[ 4x + 6x + 3 - 10 = 10x - 7 \quad \text{(Not equivalent)} \]
- \( 4x + 12 - 10 + 6x \): We have already simplified this earlier: \[ 4x + 6x + 12 - 10 = 10x + 2 \quad \text{(Equivalent)} \]
Thus, the expressions equivalent to \( 4(x + 3) - 10 + 6x \) are:
- \( 2(5x + 1) \)
- \( 4x + 12 - 10 + 6x \)
So, the final answer is:
- \( 2(5x + 1) \)
- \( 4x + 12 - 10 + 6x \)