To solve the system of equations using the substitution method, let's start with the given equations:
- \(2.5x + y = -2\) (Equation 1)
- \(3x + 2y = 0\) (Equation 2)
First, we can solve Equation 1 for \(y\):
\[ y = -2 - 2.5x \]
Now that we have \(y\) in terms of \(x\), we can substitute this expression for \(y\) into Equation 2:
\[ 3x + 2(-2 - 2.5x) = 0 \]
Now simplify and solve for \(x\):
\[ 3x - 4 - 5x = 0 \] \[ 3x - 5x - 4 = 0 \] \[ -2x - 4 = 0 \] \[ -2x = 4 \] \[ x = -2 \]
Now that we have \(x\), we can substitute back into the equation we found for \(y\):
\[ y = -2 - 2.5(-2) \] \[ y = -2 + 5 = 3 \]
So, we find the solution to the system of equations is:
\[ (x, y) = (-2, 3) \]
The system of equations has one unique solution at the point \((-2, 3)\).
The correct response is:
(−2, 3)