`(x-9)/(x+y)=(1)/(2)`
`(x)/(y)-1=3`
Bsed on the system of equations above, what is the value of x+y?

Options

A. 18
B. 24
C. 27
D. 30

Answer

To solve the system of equations given by

\[ \frac{x-9}{x+y} = \frac{1}{2} \]

and

\[ \frac{x}{y} - 1 = 3, \]

we will follow these steps:

### Step 1: Solve the first equation
Start with the first equation:

\[ \frac{x-9}{x+y} = \frac{1}{2} \]

Cross multiply to eliminate the fraction:

\[ 2(x - 9) = 1(x + y) \]

This simplifies to:

\[ 2x - 18 = x + y \]

Rearranging gives:

\[ 2x - x - y = 18 \]

Thus, we have:

\[ x - y = 18 \quad \text{(Equation 1)} \]

### Step 2: Solve the second equation
Now, consider the second equation:

\[ \frac{x}{y} - 1 = 3 \]

Add 1 to both sides:

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

Cross multiply to eliminate the fraction:

\[ x = 4y \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 2 into Equation 1
Now substitute \(x\) from Equation 2 into Equation 1:

\[ 4y - y = 18 \]

This simplifies to:

\[ 3y = 18 \]

Dividing both sides by 3 gives:

\[ y = 6 \]

### Step 4: Find the value of x
Now substitute \(y = 6\) back into Equation 2:

\[ x = 4(6) = 24 \]

### Step 5: Find \(x + y\)
Now, we can find \(x + y\):

\[ x + y = 24 + 6 = 30 \]

Thus, the value of \(x + y\) is:

\[ \boxed{30} \]