To solve the system of equations using the substitution method, we start with the two equations given:
- \( x = y - 4 \)
- \( x + 8y = 2 \)
Now we can substitute the expression for \( x \) from the first equation into the second equation.
Substituting \( x \): \[ (y - 4) + 8y = 2 \]
Now simplify the equation: \[ y - 4 + 8y = 2 \] \[ 9y - 4 = 2 \] \[ 9y = 2 + 4 \] \[ 9y = 6 \] \[ y = \frac{6}{9} = \frac{2}{3} \]
Now that we have the value of \( y \), we can substitute it back into the first equation to find \( x \): \[ x = y - 4 \] \[ x = \frac{2}{3} - 4 \] To subtract, we convert \( 4 \) to a fraction with 3 as the denominator: \[ 4 = \frac{12}{3} \] Thus: \[ x = \frac{2}{3} - \frac{12}{3} = -\frac{10}{3} \]
Now we have the solution: \[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
This does not match any of the given answer choices exactly. However, the first choice looks like an incorrect representation of fractional answers, so let’s check them more carefully.
- \( (-\frac{10}{3}, \frac{2}{3}) \) simplifies to \( (-3\frac{1}{3}, \frac{2}{3}) \), which approximates \( (-3.33, 0.67) \).
Now, to match it with the given responses:
- The first response \( (-313,23) \) looks like an incorrect formatting of \( (-\frac{10}{3}, \frac{2}{3}) \).
- The second response \( (\frac{2}{3},-313) \) does not match.
- The third \( no solution \) is incorrect since we found a solution.
- The fourth \( infinite solutions \) is also incorrect.
Thus, the solution to the system of equations is: \[ \left( -\frac{10}{3}, \frac{2}{3} \right) \]
That looks closest to the formatting of \( (-313,23) \).
Final answer: \[ \left( -\frac{10}{3}, \frac{2}{3} \right) \text{ which corresponds to } (-3\frac{1}{3}, \frac{2}{3}). \]
No direct matching choice listed seems correct, confirming this system has only one solution.
1 answer