Question

Use the substitution method to solve the system x=y−4 , x+8y=2 .(1 point)%0D%0AResponses%0D%0A%0D%0Ainfinite solutions%0D%0Ainfinite solutions%0D%0A%0D%0A(23,−313)%0D%0Aleft parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis%0D%0A%0D%0Ano solution%0D%0Ano solution%0D%0A%0D%0A(−313,23)

Answer 1

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

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

### Step 1: Substitute Equation 1 into Equation 2

We can substitute Equation 1 into Equation 2.

Replace \( x \) in the second equation:

\[
(y - 4) + 8y = 2
\]

### Step 2: Simplify the equation

Now we simplify the equation:

\[
y - 4 + 8y = 2
\]

Combine like terms:

\[
9y - 4 = 2
\]

### Step 3: Solve for \(y\)

Add 4 to both sides:

\[
9y = 6
\]

Now, divide by 9:

\[
y = \frac{6}{9} = \frac{2}{3}
\]

### Step 4: Substitute \( y \) back to find \( x \)

Now that we have \( y \), substitute \( y = \frac{2}{3} \) back into Equation 1 to find \( x \):

\[
x = y - 4 = \frac{2}{3} - 4 = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]

### Conclusion

Now we have both \( x \) and \( y \):

\[
x = -\frac{10}{3}, \quad y = \frac{2}{3}
\]

Thus, the solution to the system of equations is:

\[
\left(-\frac{10}{3}, \frac{2}{3}\right)
\]

None of the options provided seem to match this solution. The closest representation in the responses could have been simplified different interpretations depending on the formatting.

Let me check the provided options again:

- infinite solutions
- (23, -313)
- left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis
- no solution
- (−313, 23)

Given the context, it appears there was a misunderstanding in reading numerals or a miscalculation since my response does not match any listed.

However, without further appropriate decimals or arrangements, I maintain my findings that the solution to the equations set yielded results above. Please check the integrity or formatting of choices to ensure we have consistency in presentation.