If (2a/m +b/n)=2 and (a/m - b/n)= 4, find the values of 'a' and 'b' respectively.

Options

A. `2m,-2n`
B. `-2n, 2m`
C. `2m,2n`
D. `-2m,2n`

Answer

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

### Step 1: Write down the equations
We have two equations:
1. \( \frac{2a}{m} + \frac{b}{n} = 2 \) (Equation 1)
2. \( \frac{a}{m} - \frac{b}{n} = 4 \) (Equation 2)

### Step 2: Substitute variables
Let:
- \( x = \frac{1}{m} \)
- \( y = \frac{1}{n} \)

Now we can rewrite the equations:
1. \( 2ax + by = 2 \) (Equation 1 becomes)
2. \( ax - by = 4 \) (Equation 2 becomes)

### Step 3: Add the equations
Now, we will add both equations to eliminate \( by \):
\[ (2ax + by) + (ax - by) = 2 + 4 \]
This simplifies to:
\[ 3ax = 6 \]

### Step 4: Solve for \( ax \)
Now, divide both sides by 3:
\[ ax = 2 \]

### Step 5: Substitute \( ax \) back into one of the equations
We can substitute \( ax = 2 \) back into Equation 1:
\[ 2(2) + by = 2 \]
This simplifies to:
\[ 4 + by = 2 \]
Now, isolate \( by \):
\[ by = 2 - 4 \]
\[ by = -2 \]

### Step 6: Solve for \( a \) and \( b \)
Now we have:
1. \( ax = 2 \)
2. \( by = -2 \)

Substituting back for \( x \) and \( y \):
- Since \( x = \frac{1}{m} \), we have \( a \cdot \frac{1}{m} = 2 \) which gives us:
\[ a = 2m \]

- Since \( y = \frac{1}{n} \), we have \( b \cdot \frac{1}{n} = -2 \) which gives us:
\[ b = -2n \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
- \( a = 2m \)
- \( b = -2n \)

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