Question

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

infinite solutions
infinite solutions

(−31/3,2/3)
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

(2/3,−31/3)

Answer 1

To solve the system of equations using the substitution method, we have the following equations:

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

**Step 1: Substitute Equation 1 into Equation 2.**

We can substitute \( x \) from Equation 1 into Equation 2:

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

**Step 2: Simplify the equation.**

Combine the terms:

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

\[
9y - 4 = 2
\]

**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 substitute \( y = \frac{2}{3} \) into Equation 1:

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

Convert \( 4 \) to a fraction with a denominator of 3:

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

**Step 5: Write the solution.**

Thus, the solution to the system of equations is:

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

However, none of the provided answers align exactly with this solution. If we check the original answer choices:

1. Infinite solutions
2. \( \left( -\frac{31}{3}, \frac{2}{3} \right) \)
3. No solution
4. \( \left( \frac{2}{3}, -\frac{31}{3} \right) \)

None of these match the computed solution \( \left( \frac{-10}{3}, \frac{2}{3} \right) \).

Thus, if we strictly look at our solution, we conclude the following:
The system has a unique solution, which is not represented in the options given.

In this case, it's important to be cautious with the matching of formats and always double-check interim steps for accuracy.

Feel free to reach out if you have further questions!