Combine Like Terms: Simplifying (4/7)x + (5/7)y + (3/4)x + (8/9)y

To solve this problem, we'll proceed as follows:

• Step 1: Group the like terms involving $x$ and $y$ separately.
• Step 2: Find a common denominator for the terms involving $x$.
• Step 3: Add the fractions to simplify the $x$ terms.
• Step 4: Find a common denominator for the terms involving $y$.
• Step 5: Add the fractions to simplify the $y$ terms.
• Step 6: Combine the simplified terms for $x$ and $y$.

Now, let's perform these steps in detail:
Step 1: Identify and group the terms:
$\left(\frac{4}{7}x + \frac{3}{4}x\right)$ and $\left(\frac{5}{7}y + \frac{8}{9}y\right)$.

Step 2: Find a common denominator for the $x$-terms:
The denominators are 7 and 4. The least common denominator (LCD) is 28.

Step 3: Add the $x$-terms:
$\frac{4}{7}x = \frac{4 \cdot 4}{28}x = \frac{16}{28}x$
$\frac{3}{4}x = \frac{3 \cdot 7}{28}x = \frac{21}{28}x$
Adding them gives $\frac{16}{28}x + \frac{21}{28}x = \frac{37}{28}x = 1\frac{9}{28}x$.

Step 4: Find a common denominator for the $y$-terms:
The denominators are 7 and 9. The LCD is 63.

Step 5: Add the $y$-terms:
$\frac{5}{7}y = \frac{5 \cdot 9}{63}y = \frac{45}{63}y$
$\frac{8}{9}y = \frac{8 \cdot 7}{63}y = \frac{56}{63}y$
Adding them gives $\frac{45}{63}y + \frac{56}{63}y = \frac{101}{63}y = 1\frac{38}{63}y$.

Step 6: Combine the simplified terms:
The final expression is $1\frac{9}{28}x + 1\frac{38}{63}y$.

Therefore, the solution to the problem is $1\frac{9}{28}x + 1\frac{38}{63}y$.