Solve the following pair of linear equations by the substitution method:
`0.2x + 0.3y =1.3`
`0.4x + 0.5y= 2.3`

Answer

To solve the given pair of linear equations using the substitution method, we will follow these steps:

### Given Equations:
1. \( 0.2x + 0.3y = 1.3 \) (Equation 1)
2. \( 0.4x + 0.5y = 2.3 \) (Equation 2)

### Step 1: Eliminate Decimals
To make calculations easier, we can eliminate the decimals by multiplying both equations by 10.

- Multiply Equation 1 by 10:
\[ 2x + 3y = 13 \quad \text{(Equation 1')} \]

- Multiply Equation 2 by 10:
\[ 4x + 5y = 23 \quad \text{(Equation 2')} \]

### Step 2: Express one variable in terms of the other
From Equation 1', we can express \( x \) in terms of \( y \):
\[ 2x + 3y = 13 \]
Rearranging gives:
\[ 2x = 13 - 3y \]
Now, divide by 2:
\[ x = \frac{13 - 3y}{2} \quad \text{(Equation 3)} \]

### Step 3: Substitute \( x \) in the second equation
Now we substitute Equation 3 into Equation 2':
\[ 4x + 5y = 23 \]
Substituting for \( x \):
\[ 4\left(\frac{13 - 3y}{2}\right) + 5y = 23 \]

### Step 4: Simplify the equation
Now simplify the equation:
\[ 2(13 - 3y) + 5y = 23 \]
This simplifies to:
\[ 26 - 6y + 5y = 23 \]
Combining like terms gives:
\[ 26 - y = 23 \]

### Step 5: Solve for \( y \)
Now, isolate \( y \):
\[ -y = 23 - 26 \]
\[ -y = -3 \]
Thus, we find:
\[ y = 3 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now substitute \( y = 3 \) back into Equation 3 to find \( x \):
\[ x = \frac{13 - 3(3)}{2} \]
Calculating gives:
\[ x = \frac{13 - 9}{2} = \frac{4}{2} = 2 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 2, \quad y = 3 \]