To solve the system of equations using the elimination method, we start with the two equations:
- \( 2x + y = 3.5 \) (Equation 1)
- \( -x + 2y = 2.5 \) (Equation 2)
First, we can aim to eliminate one of the variables. Let's eliminate \( y \) by manipulating the two equations. We can do this by multiplying Equation 1 by 2 so that the coefficient of \( y \) in both equations will match:
\[ 2(2x + y) = 2(3.5) \] This simplifies to: \[ 4x + 2y = 7 \quad \text{(Equation 3)} \]
Now, we can write our modified system of equations:
- \( 4x + 2y = 7 \) (Equation 3)
- \( -x + 2y = 2.5 \) (Equation 2)
Next, we will subtract Equation 2 from Equation 3 to eliminate \( y \):
\[ (4x + 2y) - (-x + 2y) = 7 - 2.5 \]
This simplifies to:
\[ 4x + 2y + x - 2y = 7 - 2.5 \]
Combining the terms, we have:
\[ 5x = 4.5 \]
Now, we can solve for \( x \):
\[ x = \frac{4.5}{5} = 0.9 \]
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We'll use Equation 1:
\[ 2x + y = 3.5 \]
Substituting \( x = 0.9 \):
\[ 2(0.9) + y = 3.5 \]
This simplifies to:
\[ 1.8 + y = 3.5 \]
Now, we can solve for \( y \):
\[ y = 3.5 - 1.8 = 1.7 \]
Thus, the solution to the system of equations is:
\[ x = 0.9, \quad y = 1.7 \]
In ordered pair form, the solution is:
\[ (0.9, 1.7) \]