Question

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

Answer 1

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

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

First, we will substitute equation 1 into equation 2:

Substituting \( x \) in equation 2:
\[
(y - 4) + 8y = 2
\]

Combine like terms:
\[
y - 4 + 8y = 2 \implies 9y - 4 = 2
\]

Now, add 4 to both sides:
\[
9y = 6
\]

Now, divide by 9:
\[
y = \frac{6}{9} = \frac{2}{3}
\]

Next, we will substitute the value of \( y \) back into equation 1 to find \( x \):
\[
x = y - 4 = \frac{2}{3} - 4
\]

Convert 4 to a fraction with the same denominator:
\[
4 = \frac{12}{3}
\]

So, we have:
\[
x = \frac{2}{3} - \frac{12}{3} = \frac{2 - 12}{3} = \frac{-10}{3}
\]

Thus, the solution to the system is:
\[
\left( -\frac{10}{3}, \frac{2}{3} \right)
\]

Based on the options provided, the correct choice corresponding to the values we found is:
\[
\left( -\frac{10}{3}, \frac{2}{3} \right)
\]