To solve the system of equations using the substitution method, we start with the two equations:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
Step 1: Substitute \( x \) in the second equation.
Using the first equation, we can substitute \( x \) in the second equation:
\[ (y - 4) + 8y = 2 \]
Step 2: Simplify the equation.
Now, combine like terms and simplify:
\[ y - 4 + 8y = 2 \ 9y - 4 = 2 \]
Step 3: Solve for \( y \).
Add 4 to both sides:
\[ 9y = 6 \]
Now, divide by 9:
\[ y = \frac{6}{9} = \frac{2}{3} \]
Step 4: Substitute \( y \) back to find \( x \).
Now that we have \( y \), we substitute back to find \( x \):
\[ x = y - 4 \ x = \frac{2}{3} - 4 \ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
Final Solution:
Thus, the solution to the system is
\[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
This corresponds to the third response option from your list:
\[ \left( -3 \frac{1}{3}, \frac{2}{3} \right) \]
So the answer is:
\[ \text{(−}3 \frac{1}{3}, \frac{2}{3}\text{)} \]
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