Question

Use the substitution method to solve the system x=y−4, x+8y=2.(1 point)
Responses

(−3 1/3,2/3)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis

no solution
no solution

infinite solutions
infinite solutions

(2/3,−3 1/3)

Answer 1

To solve the system of equations using the substitution method, we start with the two equations:

1. \( x = y - 4 \)
2. \( x + 8y = 2 \)

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.