`{:((x)/(2) + y = 0.8),((7)/(x + (y)/(2)) = 10):}`

Answer

To solve the given system of equations:

1. \(\frac{x}{2} + y = 0.8\) (Equation 1)
2. \(\frac{7}{x + \frac{y}{2}} = 10\) (Equation 2)

Let's follow the steps to find the values of \(x\) and \(y\).

### Step 1: Rewrite the equations
We start with the two equations:
1. \(\frac{x}{2} + y = 0.8\)
2. \(\frac{7}{x + \frac{y}{2}} = 10\)

### Step 2: Simplify Equation 2
To simplify Equation 2, we can multiply both sides by \((x + \frac{y}{2})\):
\[ 7 = 10 \left(x + \frac{y}{2}\right) \]
Expanding this gives:
\[ 7 = 10x + 5y \]
Rearranging this, we get:
\[ 10x + 5y = 7 \quad \text{(Equation 2)} \]

### Step 3: Multiply Equation 1 by 10
Next, we can multiply Equation 1 by 10 to eliminate the fraction:
\[ 10 \left(\frac{x}{2} + y\right) = 10 \cdot 0.8 \]
This simplifies to:
\[ 5x + 10y = 8 \quad \text{(Equation 1)} \]

### Step 4: Set up the system of equations
Now we have the following system of equations:
1. \(5x + 10y = 8\) (Equation 1)
2. \(10x + 5y = 7\) (Equation 2)

### Step 5: Solve the system of equations
We can solve these equations by elimination. Let's multiply Equation 1 by 2:
\[ 2(5x + 10y) = 2 \cdot 8 \]
This gives us:
\[ 10x + 20y = 16 \quad \text{(Equation 3)} \]

Now we have:
1. \(10x + 20y = 16\) (Equation 3)
2. \(10x + 5y = 7\) (Equation 2)

### Step 6: Subtract Equation 2 from Equation 3
Now we subtract Equation 2 from Equation 3:
\[ (10x + 20y) - (10x + 5y) = 16 - 7 \]
This simplifies to:
\[ 15y = 9 \]
Dividing both sides by 15 gives:
\[ y = \frac{9}{15} = \frac{3}{5} = 0.6 \]

### Step 7: Substitute \(y\) back to find \(x\)
Now we substitute \(y = \frac{3}{5}\) back into Equation 1:
\[ 5x + 10\left(\frac{3}{5}\right) = 8 \]
This simplifies to:
\[ 5x + 6 = 8 \]
Subtracting 6 from both sides gives:
\[ 5x = 2 \]
Dividing both sides by 5 gives:
\[ x = \frac{2}{5} = 0.4 \]

### Final Answer
Thus, the solution to the system of equations is:
\[ x = 0.4, \quad y = 0.6 \]