Let's analyze the given unfinished equation:
\[ 12(x - 3) + 18 = ____ \]
First, simplify the left-hand side:
\[ 12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18 \]
Now, we can write the complete equation as:
\[ 12x - 18 = y \]
where \( y \) is the expression on the right-hand side from the options provided.
Next, we will match each expression with the number of solutions based on how they can be simplified:
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For \( 6(2x-3) \): \[ 6(2x - 3) = 12x - 18 \] Here, both sides are equal, so there are infinite solutions.
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For \( 4(3x-3) \): \[ 4(3x - 3) = 12x - 12 \] This simplifies to: \[ 12x - 18 = 12x - 12 \implies -18 = -12 \] This is a contradiction, which means there are no solutions.
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For \( 4(2x-3) \): \[ 4(2x - 3) = 8x - 12 \] This simplifies to: \[ 12x - 18 = 8x - 12 \] Rearranging gives: \[ 12x - 8x = -12 + 18 \implies 4x = 6 \implies x = \frac{3}{2} \] This gives us one solution.
So, the matches are:
- \( 6(2x-3) \): infinite solutions
- \( 4(3x-3) \): no solution
- \( 4(2x-3) \): one solution
1 answer