To solve the system of equations using the elimination method, we start with the given equations:
- \( 0.1x + 0.3y = 0.9 \)
- \( 0.1x - 0.3y = 0.2 \)
We can eliminate \( x \) by subtracting the second equation from the first:
\[ (0.1x + 0.3y) - (0.1x - 0.3y) = 0.9 - 0.2 \]
This simplifies to:
\[ 0.1x + 0.3y - 0.1x + 0.3y = 0.7 \]
Combining like terms yields:
\[ 0.6y = 0.7 \]
Now, we can solve for \( y \):
\[ y = \frac{0.7}{0.6} = \frac{7}{6} \approx 1.1667 \]
Next, we substitute \( y \) back into one of the original equations to solve for \( x \). We'll use the first equation:
\[ 0.1x + 0.3\left(\frac{7}{6}\right) = 0.9 \]
Calculating \( 0.3 \times \frac{7}{6} \):
\[ 0.3 \times \frac{7}{6} = \frac{2.1}{6} = \frac{7}{20} \]
Therefore, we substitute this back into the first equation:
\[ 0.1x + \frac{7}{20} = 0.9 \]
Now, we need to isolate \( x \):
\[ 0.1x = 0.9 - \frac{7}{20} \]
To perform this subtraction, convert 0.9 to a fraction with a denominator of 20:
\[ 0.9 = \frac{18}{20} \]
So:
\[ 0.1x = \frac{18}{20} - \frac{7}{20} = \frac{11}{20} \]
Dividing both sides by 0.1 (or multiplying by 10):
\[ x = \frac{11}{20} \times 10 = \frac{110}{20} = \frac{11}{2} = 5.5 \]
Thus, the solution to the system of equations is:
\[ x = 5.5, \quad y = \frac{7}{6} \approx 1.1667 \]
In conclusion, the solution is:
\[ \boxed{(5.5, \frac{7}{6})} \]
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