To solve the system of equations given by:
- \( y = -2x - 7 \) (Equation 1)
- \( 2y - x = 1 \) (Equation 2)
We can use substitution since Equation 1 already expresses \( y \) in terms of \( x \).
Step 1: Substitute Equation 1 into Equation 2
Substituting \( y = -2x - 7 \) into Equation 2:
\[ 2(-2x - 7) - x = 1 \]
Step 2: Simplify the equation
Distributing the 2:
\[ -4x - 14 - x = 1 \]
Combining like terms:
\[ -5x - 14 = 1 \]
Step 3: Solve for \( x \)
Add 14 to both sides:
\[ -5x = 1 + 14 \] \[ -5x = 15 \]
Now, divide by -5:
\[ x = -3 \]
Step 4: Find \( y \)
Now that we have \( x \), we can substitute back into Equation 1 to find \( y \):
\[ y = -2(-3) - 7 \] \[ y = 6 - 7 \] \[ y = -1 \]
Conclusion
The solution to the system of equations is:
\[ (x, y) = (-3, -1) \]
You can verify this solution by substituting both \( x \) and \( y \) back into Equation 2:
\[ 2(-1) - (-3) = 1 \ -2 + 3 = 1 \ 1 = 1 \]
Thus, the solution is confirmed as correct.