To solve the system of equations using the substitution method, we have:
- \( x = y - 4 \) (from the first equation)
- \( x + 8y = 2 \) (from the second equation)
We will substitute the expression for \( x \) from the first equation into the second equation:
Step 1: Substitute \( x \) in the second equation.
\[ (y - 4) + 8y = 2 \]
Step 2: Simplify the equation.
Combine like terms:
\[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \]
Step 3: Solve for \( y \).
Add 4 to both sides:
\[ 9y = 2 + 4 \] \[ 9y = 6 \]
Now, divide by 9:
\[ y = \frac{6}{9} = \frac{2}{3} \]
Step 4: Substitute \( y \) back to find \( x \).
Substitute \( y = \frac{2}{3} \) into the first equation:
\[ x = \left( \frac{2}{3} \right) - 4 \] \[ x = \frac{2}{3} - \frac{12}{3} \] \[ x = \frac{2 - 12}{3} \] \[ x = \frac{-10}{3} \]
Final Solution:
So the solution to the system is:
\[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
Verifying with the options you provided, the correct response matches:
Final Answer: \(-\frac{10}{3}, \frac{2}{3}\) which corresponds to the option \( \left( -\frac{10}{3}, \frac{2}{3} \right) \). Unfortunately, this specific option does not seem to be among the choices you provided.
So, the system appears to have a unique solution given by the point \( \left( -\frac{10}{3}, \frac{2}{3} \right) \).
1 answer