To solve the system of equations using the substitution method, we start with the given equations:
- \( x = y - 4 \) (Equation 1)
- \( x + 8y = 2 \) (Equation 2)
We can substitute Equation 1 into Equation 2.
Replace \( x \) in Equation 2 with \( y - 4 \):
\[ (y - 4) + 8y = 2 \]
Now, simplify this equation:
\[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \]
Next, add 4 to both sides of the equation:
\[ 9y = 6 \]
Now, divide by 9:
\[ y = \frac{6}{9} = \frac{2}{3} \]
Now that we have the value of \( y \), we can substitute it back into Equation 1 to find \( x \):
\[ x = y - 4 = \frac{2}{3} - 4 \]
To perform the subtraction, we convert 4 to a fraction:
\[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
Thus, we have:
\[ x = -\frac{10}{3}, \quad y = \frac{2}{3} \]
So the solution to the system of equations is:
\[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
Looking through the options provided:
- \( \left( -3 \frac{1}{3}, \frac{2}{3} \right) \) corresponds to \( (-\frac{10}{3}, \frac{2}{3}) \).
The correct answer is:
\[ \left( -3 \frac{1}{3}, \frac{2}{3} \right) \]