Solve for x and y :

`x+(y)/(4)=11," "(5x)/(6)-(y)/(3)=7`

Answer

To solve the system of equations given by:

1. \( x + \frac{y}{4} = 11 \) (Equation 1)
2. \( \frac{5x}{6} - \frac{y}{3} = 7 \) (Equation 2)

We will use the elimination method. Let's go through the steps:

### Step 1: Eliminate a variable
We will first eliminate \( y \) from the equations. To do this, we can manipulate the equations to have the same coefficient for \( y \).

Multiply Equation 1 by 3 to make the coefficient of \( y \) in Equation 1 equal to that in Equation 2:

\[ 3 \left( x + \frac{y}{4} \right) = 3 \cdot 11 \]

This gives us:

\[ 3x + \frac{3y}{4} = 33 \quad \text{(Equation 3)} \]

### Step 2: Rewrite Equation 2
Now we rewrite Equation 2 to have the same coefficient for \( y \):

\[ \frac{5x}{6} - \frac{y}{3} = 7 \]

Multiply Equation 2 by 4 to eliminate the fraction:

\[ 4 \left( \frac{5x}{6} - \frac{y}{3} \right) = 4 \cdot 7 \]

This gives us:

\[ \frac{20x}{6} - \frac{4y}{3} = 28 \]

To simplify, multiply through by 6 to eliminate the fractions:

\[ 20x - 8y = 168 \quad \text{(Equation 4)} \]

### Step 3: Align the equations
Now we have:

- Equation 3: \( 3x + \frac{3y}{4} = 33 \)
- Equation 4: \( 20x - 8y = 168 \)

To eliminate \( y \), we can multiply Equation 3 by 8:

\[ 8(3x + \frac{3y}{4}) = 8 \cdot 33 \]

This gives us:

\[ 24x + 6y = 264 \quad \text{(Equation 5)} \]

### Step 4: Solve the equations
Now we have:

- Equation 5: \( 24x + 6y = 264 \)
- Equation 4: \( 20x - 8y = 168 \)

We can now add Equation 4 and Equation 5 to eliminate \( y \):

\[ (24x + 6y) + (20x - 8y) = 264 + 168 \]

This simplifies to:

\[ 44x - 2y = 432 \]

### Step 5: Solve for \( x \)
Now, we can isolate \( x \):

\[ 44x = 432 + 2y \]

Now, let's substitute \( y \) back in from one of the earlier equations. From Equation 1, we can express \( y \):

\[ y = 4(11 - x) \]

Substituting this into the equation for \( x \):

\[ 44x = 432 + 2(4(11 - x)) \]

Simplifying gives:

\[ 44x = 432 + 8(11 - x) \]

\[ 44x = 432 + 88 - 8x \]

Combining like terms:

\[ 44x + 8x = 520 \]

\[ 52x = 520 \]

Dividing by 52 gives:

\[ x = 10 \]

### Step 6: Solve for \( y \)
Now substitute \( x = 10 \) back into Equation 1 to find \( y \):

\[ 10 + \frac{y}{4} = 11 \]

Subtracting 10 from both sides:

\[ \frac{y}{4} = 1 \]

Multiplying both sides by 4 gives:

\[ y = 4 \]

### Final Answer
Thus, the solution is:

\[ x = 10, \quad y = 4 \]