To solve the system of equations using the elimination method, we'll start with the given equations:
- \(-x - 5y = -19\)
- \(9x + y = -5\)
First, we can rearrange the first equation to express it in a more standard form:
\[ -x - 5y + 19 = 0 \quad \Rightarrow \quad x + 5y = 19 \]
Now we can use the two equations:
- \(x + 5y = 19\) (Equation 1)
- \(9x + y = -5\) (Equation 2)
Next, we'll manipulate these equations to eliminate one of the variables. Let’s eliminate \(y\) by multiplying Equation 2 by 5:
\[ 5(9x + y) = 5(-5) \quad \Rightarrow \quad 45x + 5y = -25 \quad \text{(Equation 3)} \]
Now we have:
- \(x + 5y = 19\) (Equation 1)
- \(45x + 5y = -25\) (Equation 3)
Next, we can subtract Equation 1 from Equation 3 to eliminate \(y\):
\[ (45x + 5y) - (x + 5y) = -25 - 19 \]
This simplifies to:
\[ (45x - x) + (5y - 5y) = -44 \] \[ 44x = -44 \]
Now divide both sides by 44:
\[ x = -1 \]
Now that we have the value of \(x\), we can substitute it back into one of the original equations to find \(y\). We can use Equation 1:
\[ -x - 5y = -19 \] Substituting \(x = -1\):
\[ -(-1) - 5y = -19 \] \[ 1 - 5y = -19 \] Subtract \(1\) from both sides:
\[ -5y = -19 - 1 \] \[ -5y = -20 \]
Now divide both sides by \(-5\):
\[ y = 4 \]
Thus, the solution to the system of equations is:
\[ x = -1, \quad y = 4 \]
So the final answer is:
\[ \boxed{(-1, 4)} \]