To solve the system of equations using the substitution method, we'll start with the equations given:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
We can substitute the expression for \( x \) from the first equation into the second equation.
Substituting \( x \) in the second equation:
\[ (y - 4) + 8y = 2 \]
Now combine like terms:
\[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \]
Next, add 4 to both sides to isolate the term with \( y \):
\[ 9y = 2 + 4 \] \[ 9y = 6 \]
Now, divide both sides by 9:
\[ y = \frac{6}{9} = \frac{2}{3} \]
Now that we have \( y \), we can substitute it back into the first equation to find \( x \):
\[ x = y - 4 = \frac{2}{3} - 4 \]
Converting 4 to a fraction with a denominator of 3:
\[ 4 = \frac{12}{3} \]
So now we have:
\[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
Thus, the solution to the system of equations is:
\[ x = \frac{-10}{3}, \quad y = \frac{2}{3} \]
So the final answer is:
\[ \left( x, y \right) = \left( \frac{-10}{3}, \frac{2}{3} \right) \]