To solve the system of equations using the substitution method, we start with the equations:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
We can substitute the expression for \( x \) from the first equation into the second equation.
Substituting \( x \) in 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 \( y \), we can substitute it back into the first equation to find \( x \):
\[ x = y - 4 = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
Thus, the solution to the system of equations is:
\[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
Now, none of the provided options match this result. Therefore, let's reevaluate the original system to ensure we did not make any mistakes.
- From the system:
- \(x = y - 4\)
- \(x + 8y = 2\)
We substitute again:
- \( (y - 4) + 8y = 2 \)
This simplifies to:
\[ y - 4 + 8y = 2 \implies 9y - 4 = 2 \implies 9y = 6 \implies y = \frac{2}{3} \]
And substituting back we get:
\[ x = y - 4 = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = -\frac{10}{3} \]
After careful review, the final calculated solution is \( \left( -\frac{10}{3}, \frac{2}{3} \right) \). However, since the processes and calculations were consistent without errors, the choice of options does not include this calculated result.
The options "no solution" and "infinite solutions" are not applicable since we found a unique solution. Therefore, it seems there may have been a misunderstanding in the initial options or problem interpretation.
So, the correct answer would be: \[ \left( -\frac{10}{3}, \frac{2}{3} \right) \] but this is not provided in the choices. If required to choose from the given, we cannot select any option correctly.