Question

How many solutions does this equation have?






9x+2/3 = 3x + 3
Responses
A no solutionno solution
B exactly one solutionexactly one solution
C exactly two solutionsexactly two solutions
D infinitely many solutions

Answer 1

To determine how many solutions the equation \( 9x + \frac{2}{3} = 3x + 3 \) has, we will first rearrange and simplify the equation.

1. 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:

\[
6x + \frac{2}{3} = 3
\]

2. 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:

\[
6x = \frac{9}{3} - \frac{2}{3} = \frac{7}{3}
\]

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**.