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

1 answer

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