To determine how many solutions there are for the given linear equation, let's start by simplifying and solving the equation step by step.
The equation given is:
\[ 3\left(\frac{x}{2} + 5\right) - 6 = \frac{9x + 18}{3} \]
First, we simplify each side.
Step 1: Simplifying the left side
Distribute the 3 on the left:
\[ 3 \cdot \frac{x}{2} + 3 \cdot 5 - 6 = \frac{3x}{2} + 15 - 6 = \frac{3x}{2} + 9 \]
So the left side simplifies to:
\[ \frac{3x}{2} + 9 \]
Step 2: Simplifying the right side
Now simplify the right side:
\[ \frac{9x + 18}{3} = \frac{9x}{3} + \frac{18}{3} = 3x + 6 \]
Step 3: Setting the simplified expressions equal
Now we have the equation:
\[ \frac{3x}{2} + 9 = 3x + 6 \]
Step 4: Eliminating the fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ 2\left(\frac{3x}{2} + 9\right) = 2(3x + 6) \]
This simplifies to:
\[ 3x + 18 = 6x + 12 \]
Step 5: Rearranging to isolate x
Next, rearranging the terms gives:
\[ 18 - 12 = 6x - 3x \]
Which simplifies to:
\[ 6 = 3x \]
Step 6: Solving for x
Divide both sides by 3:
\[ x = 2 \]
Conclusion
Since we found one value for \(x\), the equation has exactly one solution.
Thus, the correct response is:
B: one solution.
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