To solve the system of equations using the elimination method, we have the following equations:
- \( 2x + y = 3.5 \) (Equation 1)
- \( -x + 2y = 2.5 \) (Equation 2)
We can manipulate these equations to eliminate one of the variables. Let's eliminate \( y \) by making the coefficients of \( y \) in both equations equal.
First, we can multiply Equation 1 by 2:
\[ 2(2x + y) = 2(3.5) \]
This gives us:
\[ 4x + 2y = 7 \quad \text{(Equation 3)} \]
Now we have the system:
- \( 4x + 2y = 7 \)
- \( -x + 2y = 2.5 \)
Next, we'll subtract Equation 4 from Equation 3:
\[ (4x + 2y) - (-x + 2y) = 7 - 2.5 \]
Simplifying this leads to:
\[ 4x + 2y + x - 2y = 4.5 \]
This simplifies to:
\[ 5x = 4.5 \]
Now, solve for \( x \):
\[ x = \frac{4.5}{5} = 0.9 \]
Now that we have the value of \( x \), we can substitute it back into one of the original equations to find \( y \). Let's 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, solve for \( y \):
\[ y = 3.5 - 1.8 = 1.7 \]
So, the solution to the system of equations is:
\[ x = 0.9, \quad y = 1.7 \]
Thus, the final answer is:
\[ (x, y) = (0.9, 1.7) \]