To determine how many solutions the equation \( 9x + \frac{2}{3} = 3x + 3 \) has, we will first rearrange and simplify the equation.
- Start by isolating all terms involving \( x \) on one side. We can do this by subtracting \( 3x \) from both sides:
\[ 9x - 3x + \frac{2}{3} = 3 \]
This simplifies to:
\[ 6x + \frac{2}{3} = 3 \]
- Next, subtract \(\frac{2}{3}\) from both sides to continue isolating \( x \):
\[ 6x = 3 - \frac{2}{3} \]
To subtract \( \frac{2}{3} \), convert \( 3 \) to a fraction with the same denominator:
\[ 3 = \frac{9}{3} \]
Now the equation becomes:
\[ 6x = \frac{9}{3} - \frac{2}{3} = \frac{7}{3} \]
- Next, divide both sides by 6 to solve for \( x \):
\[ x = \frac{7}{3} \cdot \frac{1}{6} = \frac{7}{18} \]
The equation yields a single solution \( x = \frac{7}{18} \).
Since we found one unique value for \( x \), the number of solutions to the equation is:
B) exactly one solution.