To determine the number of solutions for the equation \(12(x - 3) + 18 = k\) for each expression on the right-hand side, we first simplify the left side.
Starting with the left-hand side:
\[ 12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18 \]
Now we have the equation:
\[ 12x - 18 = k \]
We can solve for \(x\):
\[ 12x = k + 18 \] \[ x = \frac{k + 18}{12} \]
The number of solutions depends on the right-hand side expression \(k\). Each expression corresponds to a specific value of \(k\):
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For \(k = 6(2x - 3)\):
\[ k = 6(2x - 3) = 12x - 18 \]
Here, both sides are equal, so the equation \(12x - 18 = 12x - 18\) holds true for all \(x\). Thus, there are infinitely many solutions.
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For \(k = 4(3x - 3)\):
\[ k = 4(3x - 3) = 12x - 12 \]
Setting up the equation:
\[ 12x - 18 = 12x - 12 \]
Simplifying gives:
\[ -18 \neq -12 \]
Since this is a false statement, there are no solutions.
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For \(k = 4(2x - 3)\):
\[ k = 4(2x - 3) = 8x - 12 \]
Setting up the equation:
\[ 12x - 18 = 8x - 12 \]
Rearranging gives:
\[ 12x - 8x = -12 + 18 \] \[ 4x = 6 \] \[ x = \frac{6}{4} = \frac{3}{2} \]
This results in a single solution. Thus, there is one solution.
Summary of solutions:
- \(6(2x-3)\): Infinitely many solutions
- \(4(3x-3)\): No solutions
- \(4(2x-3)\): One solution
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