Question

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

(−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

(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:

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

using the substitution method, we will substitute the expression for \(x\) from the first equation into the second equation.

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

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

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

### Step 2: Combine like terms

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

### 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 \(y\), substitute it back into the first equation to find \(x\):

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

### Step 5: The solution

Thus, the solution to the system is:

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

### Summary of Answers

None of the provided responses match this solution. The closest representation might be requiring a more accurate description from the options. The exact solution is \(\left( -\frac{10}{3}, \frac{2}{3} \right)\), which we can denote as:

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

Thus, we may conclude it as:

\[
\left(-\frac{10}{3}, \frac{2}{3}\right) \text{ or } (-3 \frac{1}{3}, \frac{2}{3})
\]

Based on the options given, it seems like there's an error in the provided responses. Therefore, the correct solution is not listed among those options.