Simplify 2/5x + 4/3y + 7/9x + 3/4y: Combining Like Terms with Fractions

To solve this problem, we'll follow these steps:

• Step 1: Identify the given expression and separate like terms.
• Step 2: Find a common denominator for terms with same variables.
• Step 3: Add the fractions for each group of like terms.
• Step 4: Simplify the expression to find the final result.

Now, let's work through each step:
Step 1: The expression is $\frac{2}{5}x + \frac{4}{3}y + \frac{7}{9}x + \frac{3}{4}y$. Group like terms together:
$(\frac{2}{5}x + \frac{7}{9}x) + (\frac{4}{3}y + \frac{3}{4}y)$.

Step 2: For $(\frac{2}{5}x + \frac{7}{9}x)$, find a common denominator for the fractions $2/5$ and $7/9$, which is 45.
Convert $\frac{2}{5}$ to $\frac{18}{45}$ and $\frac{7}{9}$ to $\frac{35}{45}$.

Step 3: Add the fractions for $x$:
$\frac{18}{45}x + \frac{35}{45}x = \frac{53}{45}x$.

For $(\frac{4}{3}y + \frac{3}{4}y)$, find a common denominator, which is 12.
Convert $\frac{4}{3}$ to $\frac{16}{12}$ and $\frac{3}{4}$ to $\frac{9}{12}$.

Add the fractions for $y$:
$\frac{16}{12}y + \frac{9}{12}y = \frac{25}{12}y$.

Step 4: Combine results to express the simplified form:
$\frac{53}{45}x + \frac{25}{12}y$.

As mixed numbers, the solution becomes:
$1\frac{8}{45}x + 2\frac{1}{12}y$.

Therefore, the solution to the problem is $1\frac{8}{45}x + 2\frac{1}{12}y$.