To solve the system of equations using the substitution method, we start with the two equations provided:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
First, we can 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, simplify the equation:
\[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \]
Next, add 4 to both sides:
\[ 9y = 2 + 4 \] \[ 9y = 6 \]
Now, divide both sides by 9:
\[ y = \frac{6}{9} = \frac{2}{3} \]
Now that we have \( y \), we can find \( x \) using the first equation:
\[ x = y - 4 \] \[ x = \frac{2}{3} - 4 \] \[ x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
So the solution to the system of equations is:
\[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
Now, let's see if this matches any of the given responses:
- The responses were:
- (23, −313)
- \( \left( \frac{2}{3}, -3 \frac{1}{3} \right) \)
- (−3 \frac{1}{3}, 23)
- \( \left( -3 \frac{1}{3}, \frac{2}{3} \right) \)
- no solution
- infinite solutions
The correct answer is \( \left( -\frac{10}{3}, \frac{2}{3} \right) \) which does not match any of the provided responses directly. So there is no appropriate response from the options given.