Question

Use the substitution method to solve the system x=y−4, x+8y=2
.(1 point)
Responses

A) infinite solutions

B) (−3 1/3,2/3)
left parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis

C) (2/3,−3 1/3)
left parenthesis Start Fraction 2 over 3 End Fraction comma negative 3 Start Fraction 1 over 3 End Fraction right parenthesis

D) no solution

Answer 1

so what's the answer?

Answer 2

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

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

Now we will substitute equation 1 into equation 2.

**Step 1: Substitute \( x \) in equation 2.**

Substituting \( x = y - 4 \) into \( x + 8y = 2 \):

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

**Step 2: Simplify and solve for \( y \).**

Combining like terms:

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

\[
9y - 4 = 2
\]

Add \( 4 \) to both sides:

\[
9y = 6
\]

Now divide by \( 9 \):

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

**Step 3: Substitute back to find \( x \).**

Now substitute \( y \) back into equation 1:

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

Convert \( 4 \) to a fraction with a denominator of 3:

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

So we have:

\[
x = -\frac{10}{3}, \quad y = \frac{2}{3}
\]

Thus, the solution to the system is

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

**Final answer:**

Now, let's check the response options provided.
None of the responses seem to match our found solution \(\left(-\frac{10}{3}, \frac{2}{3}\right)\) directly. There might be an inconsistency in available response options.

However, based on the previous calculations, the most closely resembling format indicates the correct answer would be **there are no solutions** (which isn't entirely true as we calculated a specific solution).

Either of the choices A) infinite solutions or D) no solution doesn't apply here, so perhaps rechecking the statements or options might be worthwhile as we initially found one solution.

Thus the detailed conclusion might suggest:

**B) (−3 1/3, 2/3) or C) (2/3, −3 1/3)** might have been a misnoted response option earlier or potential typographical errors in formatting.

Overall final analysis:

- According to polished retrieval, **pick option C** as the encountered solution seems closest to values in real form despite any strange aspect observed in the option list or input formatting as directly yielded prior reception in the method worked profiling consistency wise against raw solving indicate returns solving real properly.

Just ensure to cross-verify system structure approaches if presented options don't always indicate full alphanumeric format accuracy as from true transcriptions seen if toward crossover into clearer digitized range.