To solve the given system of equations using the elimination method, we have:
1. \( 0.1x + 0.3y = 0.9 \)
2. \( 0.1x - 0.3y = 0.2 \)
Let's label the equations for easier reference:
Equation 1: \( 0.1x + 0.3y = 0.9 \)
Equation 2: \( 0.1x - 0.3y = 0.2 \)
**Step 1: Eliminate one variable**
We can subtract Equation 2 from Equation 1 to eliminate \( x \):
\[
(0.1x + 0.3y) - (0.1x - 0.3y) = 0.9 - 0.2
\]
This simplifies to:
\[
0.3y + 0.3y = 0.7
\]
Combining like terms, we have:
\[
0.6y = 0.7
\]
**Step 2: Solve for \( y \)**
Now, divide both sides by 0.6:
\[
y = \frac{0.7}{0.6} = \frac{7}{6}
\]
**Step 3: Substitute back to find \( x \)**
Now we substitute \( y = \frac{7}{6} \) back into one of the original equations. We'll use Equation 1:
\[
0.1x + 0.3(\frac{7}{6}) = 0.9
\]
Calculating \( 0.3 \times \frac{7}{6} \):
\[
0.3 \times \frac{7}{6} = \frac{21}{60} = \frac{7}{20}
\]
Now substitute back:
\[
0.1x + \frac{7}{20} = 0.9
\]
Next, we convert \( 0.9 \) to a fraction:
\[
0.9 = \frac{9}{10} = \frac{18}{20}
\]
Now we have:
\[
0.1x + \frac{7}{20} = \frac{18}{20}
\]
Subtract \( \frac{7}{20} \) from both sides:
\[
0.1x = \frac{18}{20} - \frac{7}{20} = \frac{11}{20}
\]
**Step 4: Solve for \( x \)**
Now we divide by 0.1:
\[
x = \frac{11}{20} \div 0.1 = \frac{11}{20} \div \frac{1}{10} = \frac{11 \times 10}{20} = \frac{110}{20} = 5.5
\]
Now we have \( x = \frac{11}{2} \) and \( y = \frac{7}{6} \).
**Final answer:**
The solution to the system of equations is:
\[
\left( \frac{11}{2}, \frac{7}{6} \right)
\]
From the answer choices provided, it matches with:
\( \left( \frac{11}{2}, \frac{7}{6} \right) \)