Solve the Two-Variable Equation: 2/3(x+y) with Variable Terms

To solve this problem, we will follow these steps:

• Step 1: Identify and expand both sides of the equation.
• Step 2: Simplify the equation.
• Step 3: Compare the coefficients of like terms in the equations.
• Step 4: Solve for the missing term.

Step 1: Start by examining the equation: $y(?)+\frac{2}{3}(x+y)=\frac{2}{3}x+2\frac{8}{9}y+5xy$.

Step 2: Simplify the left side of the equation:
$\frac{2}{3}(x+y) = \frac{2}{3}x + \frac{2}{3}y$.

Step 3: Equating both sides:
$y(?) + \frac{2}{3}x + \frac{2}{3}y = \frac{2}{3}x + 2\frac{8}{9}y + 5xy$.

Step 4: Compare coefficients of like terms.

• The term $x$ on both sides already agrees with $\frac{2}{3}x$.
• The coefficient of $y$ on the left side is $\frac{2}{3}$ and on the right side, it's $2\frac{8}{9} = \frac{26}{9}$.
• The extra part in the right needs to be balanced by $y(?)$.

Step 5: Solve for the missing term by comparing coefficients:
$y(?) + \frac{2}{3}y = \frac{26}{9}y + 5xy$.

The difference to balance the $y$ terms is $2\frac{2}{9}y = \frac{26}{9}y - \frac{2}{3}y$.

The remaining term on the right side, after matching is $5xy$, can be on the left as part of $y(?).$

Therefore, the missing term $y(?)$ is equal to $2\frac{2}{9} + 5x$.

Thus, the solution to the problem is $\boxed{2\frac{2}{9} + 5x}$.