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

(−313,23)
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

infinite solutions

Answer 1

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

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

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

Using the first equation, we can substitute \( x \) in the second equation:

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

**Step 2: Simplify the equation.**

Now, combine like terms and simplify:

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

**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 that we have \( y \), we substitute back to find \( x \):

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

**Final Solution:**

Thus, the solution to the system is

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

This corresponds to the third response option from your list:

\[
\left( -3 \frac{1}{3}, \frac{2}{3} \right)
\]

So the answer is:

\[
\text{(−}3 \frac{1}{3}, \frac{2}{3}\text{)}
\]