Solve the Complex Fraction Equation: 2x(2/3x+4/7y)+6/7x(4+☐)=1⅓x²+3⅜x+2xy

To solve this equation, our goal is to determine the placeholder $☐$. Let's follow these steps:
Step 1: Expand both sides.

The left side of the equation becomes:
$\begin{aligned} 2x\left(\frac{2}{3}x + \frac{4}{7}y\right) + \frac{6}{7}x(4+☐) &= 2x \cdot \frac{2}{3}x + 2x \cdot \frac{4}{7}y + \frac{6}{7}x \cdot 4 + \frac{6}{7}x \cdot ☐ \\ &= \frac{4}{3}x^2 + \frac{8}{7}xy + \frac{24}{7}x + \frac{6}{7}x \cdot ☐. \end{aligned}$

Step 2: Simplify the right side.
The right side is already in simplified form: $1\frac{1}{3}x^2 + 3\frac{3}{7}x + 2xy = \frac{4}{3}x^2 + \frac{24}{7}x + 2xy.$

Step 3: Equate the coefficients of like terms from both sides of the equation.

• Compare $x^2$ coefficients: $\frac{4}{3} = \frac{4}{3}$

• Compare $x$ coefficients: $\frac{24}{7} = \frac{24}{7}$

• Compare $xy$ coefficients: $\frac{8}{7} + \frac{6}{7}☐ = 2$

Step 4: Solve for the placeholder $☐$ using the following $xy$ equation:

$\begin{aligned} \frac{8}{7} + \frac{6}{7}☐ &= 2 \\ \frac{6}{7}☐ &= 2 - \frac{8}{7} \\ \frac{6}{7}☐ &= \frac{14}{7} - \frac{8}{7} \\ \frac{6}{7}☐ &= \frac{6}{7} \\ ☐ &= y \end{aligned}$

Therefore, the placeholder $☐$ must be filled with $\mathbf{y}$ for the equation to be correct.