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

infinite solutions
infinite solutions

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
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Answer 1

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

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

We can substitute Equation 1 into Equation 2.

Replace \( x \) in Equation 2 with \( y - 4 \):

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

Now, simplify this equation:

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

Next, add 4 to both sides of the equation:

\[
9y = 6
\]

Now, divide by 9:

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

Now that we have the value of \( y \), we can substitute it back into Equation 1 to find \( x \):

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

To perform the subtraction, we convert 4 to a fraction:

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

Thus, we have:

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

So the solution to the system of equations is:

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

Looking through the options provided:

- \( \left( -3 \frac{1}{3}, \frac{2}{3} \right) \) corresponds to \( (-\frac{10}{3}, \frac{2}{3}) \).

The correct answer is:

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