To solve the system of equations using the substitution method, we have the following equations:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
Let's substitute the expression for \( x \) from the first equation into the second equation.
Substituting \( x = y - 4 \) into the second equation:
\[ (y - 4) + 8y = 2 \]
Now, combine like terms:
\[ y - 4 + 8y = 2 \]
\[ 9y - 4 = 2 \]
Next, add 4 to both sides:
\[ 9y = 6 \]
Now, divide by 9:
\[ y = \frac{6}{9} = \frac{2}{3} \]
Now that we have found \( y \), we can substitute this value back into the first equation to find \( x \):
\[ x = y - 4 = \frac{2}{3} - 4 \]
To subtract 4, we convert 4 into a fraction with a denominator of 3:
\[ 4 = \frac{12}{3} \]
Now subtract:
\[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
So the solution to the system of equations is:
\[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
Comparing this result with the provided options, we don’t have an exact match based on the answer possibilities provided (there seems to be some discrepancies).
From the results shown and possible choices, none of the provided solutions correctly match our derived solution:
- No solution
- Infinite solutions
- \((\frac{2}{3}, -\frac{10}{3})\)
- \((- \frac{10}{3}, \frac{2}{3})\)
The choice closest to our calculation (which seems to be misrepresented) is:
\((- \frac{10}{3}, \frac{2}{3})\).
Please ensure the options given are correct as it seems there is a misalignment. The answer found is:
\[ \left( -\frac{10}{3}, \frac{2}{3} \right). \]