To solve the system of equations using the substitution method, we have the following:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
We start by substituting \( x \) from the first equation into the second equation.
Substituting \( x \) in the second equation:
\[ (y - 4) + 8y = 2 \]
Now, simplify the equation:
\[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \] \[ 9y = 2 + 4 \] \[ 9y = 6 \] \[ y = \frac{6}{9} = \frac{2}{3} \]
Now that we have \( y \), we can find \( x \) using the first equation:
\[ x = y - 4 = \frac{2}{3} - 4 \]
To subtract 4, we convert it to a fraction with a denominator of 3:
\[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
Now, we have the solution:
\[ (x, y) = \left(\frac{-10}{3}, \frac{2}{3}\right) \]
This gives the solution for the system of equations. When looking at the multiple-choice options:
A. \((-3 \frac{1}{3}, \frac{2}{3})\)
B. \((\frac{2}{3}, -3 \frac{1}{3})\)
C. No solutions
D. Infinite solutions
None of the options matches with our calculated solution of \(\left(\frac{-10}{3}, \frac{2}{3}\right)\). Therefore, we need to check if there's been a mistake or clarify the solutions.
The calculated values do not match, indicating potential oversight with calculations or observations of expected solutions. Since equations are linear and they intersect only once, there is 1 unique solution.
Thus, technically, there should be a point for the solution that isn't found in the given choices (the system may need revalidation if a -3 or improper fraction options could represent). In this case:
Final conclusion:
The answer based on the equations without the exact match from options: Unique solution exists, but not represented in the choices. Thus might indicate rounding or non-inclusion instead of C (no solutions) or D (infinite solutions).
Correct answer (not listed):
\(\left(\frac{-10}{3}, \frac{2}{3}\right)\) or simplified representation necessary for format clarity.
If keeping to options is key though, C (no solutions) seems miscalculated. Given option “infinite” based on linear approach representation might be traced as incorrect leading into a mismatch retake.
Thus acknowledgment exists for presence but possible misalignment with expected figures.