To solve the system of equations using the substitution method, we can start with the equations given:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
Since the first equation gives us \( x \) in terms of \( y \), we can substitute \( x \) in 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 \( y \), we can substitute it back into the first equation to find \( x \):
\[ x = y - 4 = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
So, the solution to the system of equations is:
\[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
In decimal form, the answer is approximately \((-3.33, 0.67)\).
Since this point is not in the provided options, let’s check the calculation once more.
From the original equations:
- \( x = y - 4 \) substitutes into:
- \( (y - 4) + 8y = 2 \)
Which simplifies correctly as shown above leading to:
\[ y = \frac{2}{3} \] \[ x = y - 4 = \frac{2}{3} - 4 = -\frac{10}{3} \]
Now review the input options:
- A. \((-3 1/3, 2/3)\) corresponds to \(-\frac{10}{3}\) which translates to \(-3.33\) roughly, and this is the correct form.
- B. \((2/3, -3 1/3)\) does not match.
- C. no solutions do not match.
- D. Infinite solutions also do not match.
Thus the correct answer is:
A. \((-3 \frac{1}{3}, \frac{2}{3})\).