To solve the system of equations using the substitution method, we have the following equations:
- \( x = y - 4 \) (Equation 1)
- \( x + 8y = 2 \) (Equation 2)
Step 1: Substitute Equation 1 into Equation 2.
We can substitute \( x \) from Equation 1 into Equation 2:
\[ (y - 4) + 8y = 2 \]
Step 2: Simplify the equation.
Combine the terms:
\[ y - 4 + 8y = 2 \]
\[ 9y - 4 = 2 \]
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 substitute \( y = \frac{2}{3} \) into Equation 1:
\[ x = \frac{2}{3} - 4 \]
Convert \( 4 \) to a fraction with a denominator of 3:
\[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
Step 5: Write the solution.
Thus, the solution to the system of equations is:
\[ \left( \frac{-10}{3}, \frac{2}{3} \right) \]
However, none of the provided answers align exactly with this solution. If we check the original answer choices:
- Infinite solutions
- \( \left( -\frac{31}{3}, \frac{2}{3} \right) \)
- No solution
- \( \left( \frac{2}{3}, -\frac{31}{3} \right) \)
None of these match the computed solution \( \left( \frac{-10}{3}, \frac{2}{3} \right) \).
Thus, if we strictly look at our solution, we conclude the following: The system has a unique solution, which is not represented in the options given.
In this case, it's important to be cautious with the matching of formats and always double-check interim steps for accuracy.
Feel free to reach out if you have further questions!