To solve the system of equations using the substitution method, we start with the two equations:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
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, combine like terms:
\[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \]
Next, we add 4 to both sides:
\[ 9y = 6 \]
Now, divide both sides by 9:
\[ y = \frac{6}{9} = \frac{2}{3} \]
Now that we have \( y \), we can substitute back to find \( x \) using the first equation:
\[ x = y - 4 = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3} \]
Thus, the solution to the system is:
\[ \left( \frac{-10}{3}, \frac{2}{3} \right) \]
However, if we look back at the provided options:
- (−3 1/3,2/3)
- no solution
- infinite solutions
- (2/3,−3 1/3)
It appears that neither of these pairs matches our results.
Let's return briefly to check the values of \( x \) and \( y \):
Calculating \( x \) again: Using \( y = \frac{2}{3} \): \[ x = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = \frac{-10}{3} \]
The pair \( \left(-\frac{10}{3}, \frac{2}{3}\right) \) isn't listed in the responses. Therefore, the closest match based on the options given might have been inaccurate in representation.
Conclusion: The system has one unique solution \( \left(-\frac{10}{3}, \frac{2}{3}\right) \), which does not match any provided option. Thus, it appears there is either an error in the options or in the transcription; but from our working solution, we conclude there is a unique solution, not infinite or none.
1 answer