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 will eliminate one of the variables by manipulating the equations.
Step 1: Make the coefficients of \(y\) the same.
To eliminate \(y\), we can multiply Equation 1 by 2 to match the coefficient of \(y\) in Equation 2:
\[ 2(2x + y) = 2(3.5) \] This gives us:
\[ 4x + 2y = 7 \quad \text{(Equation 3)} \]
Step 2: Now, we can set up the system with Equation 3 and Equation 2:
- \( 4x + 2y = 7 \) (Equation 3)
- \( -x + 2y = 2.5 \) (Equation 2)
Step 3: Eliminate \(y\).
We can subtract Equation 2 from Equation 3 to eliminate \(y\):
\[ (4x + 2y) - (-x + 2y) = 7 - 2.5 \]
This simplifies to:
\[ 4x + 2y + x - 2y = 4.5 \] \[ 5x = 4.5 \]
Step 4: Solve for \(x\).
Dividing both sides by 5:
\[ x = \frac{4.5}{5} = 0.9 \]
Step 5: Substitute \(x\) back into one of the original equations to find \(y\).
We will use Equation 1:
\[ 2(0.9) + y = 3.5 \] \[ 1.8 + y = 3.5 \] \[ y = 3.5 - 1.8 \] \[ y = 1.7 \]
Solution
Thus, the solution to the system of equations is:
\[ x = 0.9, \quad y = 1.7 \]
So, the solution is \( (0.9, 1.7) \).