Question

Use the substitution method to solve the system x = y - 4, x + 8y = 2.

A: ( 2/3, -3 1/3)

B: ( -3 1/3, 2/3)

C: Infinite solutions

D: No solutions

Answer 1

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

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

We can substitute the expression for \( x \) from the first equation into the second equation.

Substituting \( x = y - 4 \) into the second equation:

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

Now, combine like terms:

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

Next, we solve for \( y \):

Add 4 to both sides:

\[
9y = 6
\]

Now divide by 9:

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

Now that we have \( y \), we can substitute it back into the first equation to find \( x \):

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

So we have:

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

Now, let's find out what these values correspond to in the answer options given:

- A: \( \left( \frac{2}{3}, -3 \frac{1}{3} \right) \) which is \( \left( \frac{2}{3}, -\frac{10}{3} \right) \) → Not a match.
- B: \( \left( -3 \frac{1}{3}, \frac{2}{3} \right) \) is \( \left( -\frac{10}{3}, \frac{2}{3} \right) \) → This is a match!
- C: Infinite solutions
- D: No solutions

Therefore, the correct answer is:

**B: \( \left( -3 \frac{1}{3}, \frac{2}{3} \right) \)**.