Question

Use the substitution method to solve the system x=y−4 , x+8y=2 .(1 point) Responses 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 (23,−313) left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis infinite solutions

Answer 1

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

1. \( x = y - 4 \) (from the first equation)
2. \( x + 8y = 2 \) (from the second equation)

We will substitute the expression for \( x \) from the first equation into the second equation:

**Step 1:** Substitute \( x \) in the second equation.

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

**Step 2:** Simplify the equation.

Combine like terms:

\[ y - 4 + 8y = 2 \]
\[ 9y - 4 = 2 \]

**Step 3:** Solve for \( y \).

Add 4 to both sides:

\[ 9y = 2 + 4 \]
\[ 9y = 6 \]

Now, divide by 9:

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

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

Substitute \( y = \frac{2}{3} \) into the first equation:

\[ x = \left( \frac{2}{3} \right) - 4 \]
\[ x = \frac{2}{3} - \frac{12}{3} \]
\[ x = \frac{2 - 12}{3} \]
\[ x = \frac{-10}{3} \]

**Final Solution:**

So the solution to the system is:

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

Verifying with the options you provided, the correct response matches:

**Final Answer:**
\(-\frac{10}{3}, \frac{2}{3}\) which corresponds to the option \( \left( -\frac{10}{3}, \frac{2}{3} \right) \). Unfortunately, this specific option does not seem to be among the choices you provided.

So, the system appears to have a unique solution given by the point \( \left( -\frac{10}{3}, \frac{2}{3} \right) \).