We are tasked with solving the equation:

$12 - \frac{y}{3} = \frac{3}{4}y + 2$

Let's solve this step by step.

Step 1: Eliminate fractions

To simplify solving, we first eliminate the fractions by finding a common denominator. The denominators in this equation are $3$ and $4$. The least common denominator (LCD) of $3$ and $4$ is $12$. Multiply every term by $12$:

$12 \cdot 12 - 12 \cdot \frac{y}{3} = 12 \cdot \frac{3}{4}y + 12 \cdot 2$

Simplify each term:

$144 - 4y = 9y + 24$

Step 2: Rearrange terms

Bring all terms involving $y$ to one side and constants to the other:

$144 - 24 = 9y + 4y$

$120 = 13y$

Step 3: Solve for $y$

Divide both sides by $13$:

$y = \frac{120}{13}$

Final Answer:

$y = \frac{120}{13} \approx 9.23 \, \text{(approximate value)}$

Would you like a more detailed breakdown of any step? Let me know!

Related Questions:

1. How do you determine the least common denominator (LCD) in equations involving fractions?
2. What are alternative methods for solving equations with fractions?
3. Can this problem be solved graphically? How?
4. What does the solution $y = \frac{120}{13}$ represent in a real-world context?
5. How do you verify the solution to an equation?

Tip:

Always double-check your solution by substituting it back into the original equation!