To solve the system of equations using elimination, we start with the following equations:
- \( 5x + 8y = -3 \) (Equation 1)
- \( -x + y = -15 \) (Equation 2)
To eliminate one of the variables, we'll first manipulate Equation 2 to align it with Equation 1. We can multiply Equation 2 by 5:
\[ 5(-x + y) = 5(-15) \]
This simplifies to:
\[ -5x + 5y = -75 \quad \text{(Equation 3)} \]
Now we have the following system:
- \( 5x + 8y = -3 \)
- \( -5x + 5y = -75 \)
Next, we will add Equation 1 and Equation 3:
\[ (5x + 8y) + (-5x + 5y) = -3 + (-75) \]
Simplifying the left side gives:
\[ 0x + (8y + 5y) = -78 \]
which simplifies to:
\[ 13y = -78 \]
Now divide both sides by 13 to solve for \( y \):
\[ y = \frac{-78}{13} = -6 \]
Now that we have \( y \), we can substitute it back into Equation 2 to solve for \( x \):
Substituting into Equation 2:
\[ -x + (-6) = -15 \]
This simplifies to:
\[ -x - 6 = -15 \]
Now add 6 to both sides:
\[ -x = -15 + 6 \]
This simplifies to:
\[ -x = -9 \]
Thus, multiplying both sides by -1, we get:
\[ x = 9 \]
Now we have the solution:
\[ x = 9 \quad \text{and } \quad y = -6 \]
To summarize, the solution to the system of equations is:
\[ (x, y) = (9, -6) \]