Solve the following equations :
`0.2x+0.3y=1.3`
`0.4x+0.5y=2.3`

Answer

To solve the equations \(0.2x + 0.3y = 1.3\) and \(0.4x + 0.5y = 2.3\), we can follow these steps:

### Step 1: Eliminate decimals
To eliminate the decimals, we can multiply both equations by 10.

1. Multiply the first equation by 10:
\[ 10(0.2x + 0.3y) = 10(1.3) \implies 2x + 3y = 13 \quad \text{(Equation 1)} \]

2. Multiply the second equation by 10:
\[ 10(0.4x + 0.5y) = 10(2.3) \implies 4x + 5y = 23 \quad \text{(Equation 2)} \]

### Step 2: Make the coefficients of \(x\) the same
To eliminate \(x\), we can make the coefficients of \(x\) in both equations the same. We can multiply Equation 1 by 2:

\[ 2(2x + 3y) = 2(13) \implies 4x + 6y = 26 \quad \text{(Equation 3)} \]

Now we have:
- Equation 2: \(4x + 5y = 23\)
- Equation 3: \(4x + 6y = 26\)

### Step 3: Subtract the equations
Now we can subtract Equation 2 from Equation 3:

\[ (4x + 6y) - (4x + 5y) = 26 - 23 \]

This simplifies to:

\[ y = 3 \]

### Step 4: Substitute \(y\) back into one of the original equations
Now that we have \(y\), we can substitute it back into Equation 1 to find \(x\):

\[ 2x + 3(3) = 13 \]

This simplifies to:

\[ 2x + 9 = 13 \]

Subtract 9 from both sides:

\[ 2x = 4 \]

Now divide by 2:

\[ x = 2 \]

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