Solve:`{:(x - 2y = - 2),(2x + y = 16):}`

Answer

To solve the system of equations given by:

1. \( x - 2y = -2 \) (Equation 1)
2. \( 2x + y = 16 \) (Equation 2)

we will use the substitution method.

### Step 1: Solve for \( x \) in terms of \( y \) from Equation 1.

Starting with Equation 1:

\[ x - 2y = -2 \]

We can isolate \( x \) by adding \( 2y \) to both sides:

\[ x = -2 + 2y \]

### Step 2: Substitute \( x \) in Equation 2.

Now that we have \( x \) in terms of \( y \), we substitute this expression into Equation 2:

\[ 2x + y = 16 \]

Substituting \( x = -2 + 2y \):

\[ 2(-2 + 2y) + y = 16 \]

### Step 3: Simplify the equation.

Distributing \( 2 \) in the equation:

\[ -4 + 4y + y = 16 \]

Combine like terms:

\[ -4 + 5y = 16 \]

### Step 4: Solve for \( y \).

Now, add \( 4 \) to both sides:

\[ 5y = 16 + 4 \]

\[ 5y = 20 \]

Now, divide both sides by \( 5 \):

\[ y = \frac{20}{5} = 4 \]

### Step 5: Substitute \( y \) back to find \( x \).

Now that we have \( y = 4 \), we can substitute this value back into the equation we derived for \( x \):

\[ x = -2 + 2y \]

Substituting \( y = 4 \):

\[ x = -2 + 2(4) \]

\[ x = -2 + 8 \]

\[ x = 6 \]

### Final Solution:

The solution to the system of equations is:

\[ x = 6, \quad y = 4 \]