`x-y=0.9; 11/(2(x+y))=10.5`

Answer

To solve the system of equations given by \( x - y = 0.9 \) and \( \frac{11}{2(x+y)} = 10.5 \), we will follow these steps:

### Step 1: Rewrite the equations
We have two equations:
1. \( x - y = 0.9 \) (Equation 1)
2. \( \frac{11}{2(x+y)} = 10.5 \) (Equation 2)

### Step 2: Simplify Equation 2
To eliminate the fraction in Equation 2, we can multiply both sides by \( 2(x+y) \):
\[ 11 = 10.5 \cdot 2(x+y) \]
This simplifies to:
\[ 11 = 21(x+y) \]
Now, divide both sides by 21:
\[ x + y = \frac{11}{21} \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations
Now we have:
1. \( x - y = 0.9 \) (Equation 1)
2. \( x + y = \frac{11}{21} \) (Equation 3)

We can add Equation 1 and Equation 3 to eliminate \( y \):
\[ (x - y) + (x + y) = 0.9 + \frac{11}{21} \]
This simplifies to:
\[ 2x = 0.9 + \frac{11}{21} \]

### Step 4: Find a common denominator
To add \( 0.9 \) and \( \frac{11}{21} \), we convert \( 0.9 \) to a fraction:
\[ 0.9 = \frac{9}{10} \]
Now, we need a common denominator for \( \frac{9}{10} \) and \( \frac{11}{21} \). The least common multiple of 10 and 21 is 210.

Convert \( \frac{9}{10} \):
\[ \frac{9}{10} = \frac{9 \times 21}{10 \times 21} = \frac{189}{210} \]

Convert \( \frac{11}{21} \):
\[ \frac{11}{21} = \frac{11 \times 10}{21 \times 10} = \frac{110}{210} \]

Now add them:
\[ 2x = \frac{189}{210} + \frac{110}{210} = \frac{299}{210} \]

### Step 5: Solve for \( x \)
Now divide both sides by 2:
\[ x = \frac{299}{420} \]

### Step 6: Substitute \( x \) back to find \( y \)
Now substitute \( x \) back into Equation 1:
\[ \frac{299}{420} - y = 0.9 \]
Convert \( 0.9 \) to a fraction:
\[ 0.9 = \frac{9}{10} = \frac{378}{420} \]
Now we have:
\[ \frac{299}{420} - y = \frac{378}{420} \]
Rearranging gives:
\[ y = \frac{299}{420} - \frac{378}{420} = -\frac{79}{420} \]

### Final Solution
Thus, the solutions are:
\[ x = \frac{299}{420}, \quad y = -\frac{79}{420} \]

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