The correct answer is A. a unique solution.
To solve a system of linear equations, we can use the elimination method. In this method, we eliminate one of the variables by adding or subtracting the equations so that the coefficients of the variable are negative inverses of each other. Then, we can solve for the remaining variable.
In this case, we can eliminate $y$ by adding the equations together. When we do this, we get the equation $6x = 13$. Solving for $x$, we get $x = \frac{13}{6}$.
Now that we know the value of $x$, we can plug it back into one of the original equations to solve for $y$. Plugging in $x = \frac{13}{6}$ into the first equation, we get the equation $\frac{2}{3} \cdot \frac{13}{6} + y = 7$. Solving for $y$, we get $y = \frac{1}{3}$.
Therefore, the system of linear equations has a unique solution, which is $(x, y) = \left(\frac{13}{6}, \frac{1}{3}\right)$.
Option B is incorrect because the system of linear equations has a solution. Option C is incorrect because the system of linear equations has a unique solution. Option D is incorrect because the system of linear equations does not have exactly two distinct solutions.