To use the substitution method to solve the system of equations \( x = y - 4 \) and \( x + 8y = 2 \), we start by substituting the expression for \( x \) in the second equation.
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Substitute \( x \) from the first equation into the second equation: \[ (y - 4) + 8y = 2 \]
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Combine like terms: \[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \]
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Add 4 to both sides: \[ 9y = 6 \]
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Divide by 9: \[ y = \frac{6}{9} = \frac{2}{3} \]
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Now substitute \( y \) back into the first equation to find \( x \): \[ x = \frac{2}{3} - 4 \] \[ x = \frac{2}{3} - \frac{12}{3} = -\frac{10}{3} \]
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The solution is: \[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
After checking the options, it seems that there was a small error interpretation in the solutions provided. Based on our work, the solution \( \left( -\frac{10}{3}, \frac{2}{3} \right) \) does not match the provided answers exactly.
Please double-check if you need to operate with different terms or if the question needs adjustments, but the calculation yields a unique solution. Thus, the conclusion is:
There is one solution: \[ \left( -\frac{10}{3}, \frac{2}{3} \right) \text{ (not listed in options)} \]
So the best interpretation of the available options given might lead to an error in the choices provided rather than the process we described. Hence, the answer should be one distinct solution, despite options reflecting differently.
Review your choices or check if the equations differed from what was presented in the beginning.