Solve for ` x and y` :
` 0.4 x - 1.5 y = 6.5 `,
` 0.3 x + 0.2 y = 0.9`.

Options

A. `x = -5` and `y = -3`
B. `x = 5` and `y = -3`
C. `x = 5` and `y = 3`
D. none of these

Explanation

To solve the system of equations:

1. **Equations Given:**
\[ 0.4x - 1.5y = 6.5 \quad \text{(1)} \]
\[ 0.3x + 0.2y = 0.9 \quad \text{(2)} \]

2. **Eliminate Decimals:**
To make calculations easier, we can eliminate the decimals by multiplying both equations by suitable factors. Let's multiply equation (1) by 10 and equation (2) by 10.

From equation (1):
\[ 10(0.4x - 1.5y) = 10(6.5) \]
This gives:
\[ 4x - 15y = 65 \quad \text{(3)} \]

From equation (2):
\[ 10(0.3x + 0.2y) = 10(0.9) \]
This gives:
\[ 3x + 2y = 9 \quad \text{(4)} \]

3. **Solve the System of Equations:**
Now we have a new system of equations:
\[ 4x - 15y = 65 \quad \text{(3)} \]
\[ 3x + 2y = 9 \quad \text{(4)} \]

We can solve for \(x\) and \(y\) using the elimination method. Let's multiply equation (4) by 15 to align the coefficients of \(y\).

\[ 15(3x + 2y) = 15(9) \]
This gives:
\[ 45x + 30y = 135 \quad \text{(5)} \]

Now, we will multiply equation (3) by 2 to align the coefficients of \(y\) as well.

\[ 2(4x - 15y) = 2(65) \]
This gives:
\[ 8x - 30y = 130 \quad \text{(6)} \]

4. **Add Equations (5) and (6):**
Now we can add equations (5) and (6):
\[ (45x + 30y) + (8x - 30y) = 135 + 130 \]
This simplifies to:
\[ 53x = 265 \]

5. **Solve for \(x\):**
Now, divide both sides by 53:
\[ x = \frac{265}{53} = 5 \]

6. **Substitute \(x\) back to find \(y\):**
Now substitute \(x = 5\) into equation (4):
\[ 3(5) + 2y = 9 \]
This simplifies to:
\[ 15 + 2y = 9 \]
Subtract 15 from both sides:
\[ 2y = 9 - 15 \]
\[ 2y = -6 \]
Divide both sides by 2:
\[ y = -3 \]

7. **Final Solution:**
The solution to the system of equations is:
\[ x = 5, \quad y = -3 \]