https://file.aiquickdraw.com/mathbot/file/1726281294109vksbroyi.jpg

The given equation in the image can be translated as:

If \( \frac{x}{y} = \frac{2}{3} \), then find the value of:

\[
\frac{3x + 5y}{3x - 5y} + \frac{6x + 4y}{6x - y}
\]

### Step 1: Use the given ratio \( \frac{x}{y} = \frac{2}{3} \)
This implies that \( x = \frac{2}{3}y \).

### Step 2: Substitute \( x = \frac{2}{3}y \) in the expression.

We will now substitute \( x = \frac{2}{3}y \) into the expression:

\[
\frac{3x + 5y}{3x - 5y} + \frac{6x + 4y}{6x - y}
\]

#### For the first term:

\[
\frac{3x + 5y}{3x - 5y}
= \frac{3 \left( \frac{2}{3}y \right) + 5y}{3 \left( \frac{2}{3}y \right) - 5y}
= \frac{2y + 5y}{2y - 5y}
= \frac{7y}{-3y}
= -\frac{7}{3}
\]

#### For the second term:

\[
\frac{6x + 4y}{6x - y}
= \frac{6 \left( \frac{2}{3}y \right) + 4y}{6 \left( \frac{2}{3}y \right) - y}
= \frac{4y + 4y}{4y - y}
= \frac{8y}{3y}
= \frac{8}{3}
\]

### Step 3: Add the two simplified terms

Now, add the two fractions:

\[
-\frac{7}{3} + \frac{8}{3} = \frac{-7 + 8}{3} = \frac{1}{3}
\]

### Final Answer:

The value of the given expression is \( \frac{1}{3} \).

---

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. How would the solution change if \( \frac{x}{y} = \frac{3}{2} \)?
2. What happens if we use \( \frac{3x + 4y}{3x - 4y} + \frac{6x + 3y}{6x - y} \)?
3. Can you express the result in decimal form for better understanding?
4. How do you simplify algebraic fractions like \( \frac{a + b}{a - b} \)?
5. How would you verify the result by cross-multiplying?

**Tip:** Always try to simplify the algebraic expressions before substituting values to avoid calculation mistakes.

If x/y = 2/3, then find the value of (3x + 5y)/(3x - 5y) + (6x + 4y)/(6x - y).

Learn how to solve the algebraic expression (3x + 5y)/(3x - 5y) + (6x + 4y)/(6x - y) given that x/y = 2/3. This step-by-step guide walks through substitution, simplification, and fraction addition. Suitable for students in Grades 9-11.

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