Solve the following system of equations :
`(2x)/(a)+(y)/(b)=2, (x)/(a)-(y)/(b)=4:`

Options

A. `(2)/(a),(2)/(a)`
B. `2a, -2b`
C. `-2a, 2b`
D. `(a)/(2), -(b)/(2)`

Answer

To solve the system of equations:

1. \(\frac{2x}{a} + \frac{y}{b} = 2\)
2. \(\frac{x}{a} - \frac{y}{b} = 4\)

we will follow these steps:

### Step 1: Substitute Variables
Let:
\[ u = \frac{x}{a} \quad \text{and} \quad v = \frac{y}{b} \]
This transforms our equations into:
1. \(2u + v = 2\) (Equation 1)
2. \(u - v = 4\) (Equation 2)

### Step 2: Solve for \(v\) in terms of \(u\)
From Equation 1:
\[ v = 2 - 2u \]

### Step 3: Substitute \(v\) into Equation 2
Substituting \(v\) into Equation 2 gives:
\[ u - (2 - 2u) = 4 \]
This simplifies to:
\[ u - 2 + 2u = 4 \]
Combining like terms:
\[ 3u - 2 = 4 \]

### Step 4: Solve for \(u\)
Add 2 to both sides:
\[ 3u = 6 \]
Now divide by 3:
\[ u = 2 \]

### Step 5: Substitute \(u\) back to find \(v\)
Now substitute \(u = 2\) back into the equation for \(v\):
\[ v = 2 - 2(2) = 2 - 4 = -2 \]

### Step 6: Substitute \(u\) and \(v\) back to find \(x\) and \(y\)
Recall the definitions of \(u\) and \(v\):
\[ u = \frac{x}{a} \Rightarrow x = 2a \]
\[ v = \frac{y}{b} \Rightarrow y = -2b \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 2a, \quad y = -2b \]