Solve the following system of equations :
`x-y=0.9,(11)/(2(x+y))=1:`

Options

A. `3.2, 3.5`
B. `3.2, 2.3`
C. `2.3, 2.5`
D. `4, 3.1`

Answer

To solve the system of equations given by:

1. \( x - y = 0.9 \)
2. \( \frac{11}{2(x+y)} = 1 \)

we can follow these steps:

### Step 1: Solve the first equation for \( x \)
From the first equation, we can express \( x \) in terms of \( y \):

\[ x = y + 0.9 \]

### Step 2: Simplify the second equation
The second equation can be rewritten by multiplying both sides by \( 2(x+y) \):

\[ 11 = 2(x+y) \]

### Step 3: Express \( x+y \) in terms of a constant
From the equation \( 11 = 2(x+y) \), we can solve for \( x+y \):

\[ x + y = \frac{11}{2} = 5.5 \]

### Step 4: Substitute \( x \) from Step 1 into the equation from Step 3
Now substitute \( x \) from Step 1 into the equation \( x + y = 5.5 \):

\[ (y + 0.9) + y = 5.5 \]

### Step 5: Combine like terms
Combine the terms involving \( y \):

\[ 2y + 0.9 = 5.5 \]

### Step 6: Isolate \( y \)
Subtract \( 0.9 \) from both sides:

\[ 2y = 5.5 - 0.9 \]

Calculating the right side:

\[ 2y = 4.6 \]

Now, divide by \( 2 \):

\[ y = \frac{4.6}{2} = 2.3 \]

### Step 7: Substitute \( y \) back to find \( x \)
Now that we have \( y \), substitute it back into the equation for \( x \):

\[ x = y + 0.9 = 2.3 + 0.9 = 3.2 \]

### Final Solution
Thus, the solution to the system of equations is:

\[ x = 3.2, \quad y = 2.3 \]

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