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'll substitute Equation 1 into Equation 2.
Step 1: Substitute Equation 1 into Equation 2
Replace \( x \) in Equation 2 with \( y - 4 \):
\[ (y - 4) + 8y = 2 \]
Step 2: Simplify the equation
Combine like terms:
\[ y - 4 + 8y = 2 \]
\[ 9y - 4 = 2 \]
Step 3: Solve for \( y \)
Add 4 to both sides:
\[ 9y = 6 \]
Now, divide both sides by 9:
\[ y = \frac{6}{9} = \frac{2}{3} \]
Step 4: Substitute \( y \) back into Equation 1 to find \( x \)
Now that we have \( y \), we substitute it back into Equation 1 to find \( x \):
\[ x = y - 4 \]
\[ x = \frac{2}{3} - 4 \]
Convert 4 into a fraction with a denominator of 3:
\[ x = \frac{2}{3} - \frac{12}{3} \]
\[ x = \frac{2 - 12}{3} = \frac{-10}{3} \]
Final Solution
Thus, the solution to the system of equations is:
\[ (x, y) = \left( -\frac{10}{3}, \frac{2}{3} \right) \]