If `(x+y)/(x-y)=5/3` and `x/((y+2))=2`, the value of (x,y) is

Options

A. (4,1)
B. (2,8)
C. (1,4)
D. (8,2)

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Set up the equations
We are given two equations:
1. \(\frac{x+y}{x-y} = \frac{5}{3}\)
2. \(\frac{x}{y+2} = 2\)

### Step 2: Cross-multiply the first equation
From the first equation, we can cross-multiply:
\[ 3(x+y) = 5(x-y) \]
Expanding both sides gives:
\[ 3x + 3y = 5x - 5y \]

### Step 3: Rearrange the equation
Now, rearranging the equation to group \(x\) and \(y\) terms:
\[ 3x + 3y - 5x + 5y = 0 \]
This simplifies to:
\[ -2x + 8y = 0 \]
or:
\[ 2x = 8y \]
Dividing both sides by 2 gives:
\[ x = 4y \quad \text{(Equation 1)} \]

### Step 4: Substitute into the second equation
Now, we take the second equation:
\[ \frac{x}{y+2} = 2 \]
Cross-multiplying gives:
\[ x = 2(y + 2) \]
Expanding this gives:
\[ x = 2y + 4 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2
Now, we substitute \(x = 4y\) from Equation 1 into Equation 2:
\[ 4y = 2y + 4 \]

### Step 6: Solve for \(y\)
Rearranging gives:
\[ 4y - 2y = 4 \]
This simplifies to:
\[ 2y = 4 \]
Dividing both sides by 2 gives:
\[ y = 2 \]

### Step 7: Find \(x\)
Now, substitute \(y = 2\) back into Equation 1:
\[ x = 4y = 4 \times 2 = 8 \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:
\[ (x, y) = (8, 2) \]