Question

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

no solution
no solution

infinite solutions
infinite solutions

(2/3,−3/13)
(
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

(−3/13,2/3)

Answer 1

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

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

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

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

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

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

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

Combine like terms:

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

Now, add 4 to both sides:

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

Now, divide both sides by 9:

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

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

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

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

To subtract, convert 4 to a fraction:

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

### Step 4: Write the solution.

Thus, the solution to the system is:

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

Looking at the response options provided, none of them match \( \left( -\frac{10}{3}, \frac{2}{3} \right) \).

It seems like the correct answer isn't listed in your provided options. Based on the calculations, you have:

### The solution is:

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