Question

Use the substitution method to solve the system x=y−4 , x+8y=2 .(1 point) Responses (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 no solution (−313,23) left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis infinite solutions

Answer 1

To solve the system of equations using the substitution method, we'll start with the two equations you provided:

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

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

Let's substitute \( x \) from Equation 1 into Equation 2:

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

### Step 2: Simplify the equation

Combine like terms:

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

This simplifies to:

\[
9y - 4 = 2
\]

### 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 \( y = \frac{2}{3} \) back into Equation 1 to find \( x \):

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

Convert 4 to a fraction with the same denominator:

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

### Step 5: Write the solution

Thus, the solution of the system is:

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

None of the provided responses match this solution directly. However, we can look at the nature of the solution to determine the appropriate response.

Since we found a unique solution, the answer from the given options is **not included**. But to address clarification:

- If \( \left(-\frac{10}{3}, \frac{2}{3}\right) \) was in the options, that would match our solution.

Given the responses you provided, there may be a misalignment with the expected solutions.

#### Final Result:
The solution is \( \left( -\frac{10}{3}, \frac{2}{3} \right) \).