Question

Solve the following system of equations using the elimination method.

2x+y=3.5
2
𝑥
+
𝑦
=
3.5
, −x+2y=2.5

Answer 1

To solve the system of equations using the elimination method, we have the following equations:

1. \( 2x + y = 3.5 \) (Equation 1)
2. \( -x + 2y = 2.5 \) (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:

1. \( 4x + 2y = 7 \) (Equation 3)
2. \( -x + 2y = 2.5 \) (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 \( (0.9, 1.7) \).