To solve the system of equations using the substitution method, we can start with the two given equations:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
Since the first equation expresses \( x \) in terms of \( y \), we can substitute this expression into the second equation.
Substituting \( x = y - 4 \) into the second equation:
\[ (y - 4) + 8y = 2 \]
Now, we will simplify this equation:
\[ y - 4 + 8y = 2 \]
Combine the \( y \) terms:
\[ 9y - 4 = 2 \]
Now add 4 to both sides:
\[ 9y = 2 + 4 \] \[ 9y = 6 \]
Now, divide both sides by 9 to solve for \( y \):
\[ y = \frac{6}{9} = \frac{2}{3} \]
Now that we have the value of \( y \), we can substitute it back into the first equation to find \( x \):
\[ x = y - 4 \] \[ x = \frac{2}{3} - 4 \] \[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
Thus, the solution to the system of equations is:
\[ x = -\frac{10}{3}, \quad y = \frac{2}{3} \]
The final answer is: \[ (x, y) = \left(-\frac{10}{3}, \frac{2}{3}\right) \]