Use the substitution method to solve the system x=y−4 , x+8y=2 .(1 point)%0D%0AResponses%0D%0A%0D%0A(−313,23)%0D%0Aleft parenthesis negative 3 Start Fraction 1 over 3 End Fraction comma Start Fraction 2 over 3 End Fraction right parenthesis%0D%0A%0D%0Ainfinite solutions%0D%0Ainfinite solutions%0D%0A%0D%0Ano solution%0D%0Ano solution%0D%0A%0D%0A(23,−313)

Answer 1

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

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

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

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

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

Now, simplify the equation:

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

Combine like terms:

\[ 9y - 4 = 2 \]

Next, add 4 to both sides:

\[ 9y = 6 \]

Now, divide both sides by 9:

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

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

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

To perform the subtraction, convert 4 into a fraction:

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

Thus, the solution to the system of equations is:

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

In decimal form, this corresponds to \( (-3.33, 0.67) \).

Looking at the options provided, it appears that none of the options exactly match this result. However, the closest matching format, if re-written, is:

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

So, the answer choice that matches this solution is:

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