To solve the system of equations using the elimination method, we start with the given equations:
- \( 0.1x + 0.3y = 0.9 \) (Equation 1)
- \( 0.1x - 0.3y = 0.2 \) (Equation 2)
First, let's eliminate one of the variables, say \( y \). We can do this by adding the two equations together.
Step 1: Add the equations together:
\[ (0.1x + 0.3y) + (0.1x - 0.3y) = 0.9 + 0.2 \]
This simplifies to:
\[ 0.2x + 0y = 1.1 \]
Thus, we have:
\[ 0.2x = 1.1 \]
Step 2: Solve for \( x \):
To solve for \( x \), divide both sides by \( 0.2 \):
\[ x = \frac{1.1}{0.2} = 5.5 \]
Step 3: Substitute \( x \) back into one of the original equations to find \( y \):
We can substitute \( x = 5.5 \) into Equation 1:
\[ 0.1(5.5) + 0.3y = 0.9 \]
Calculating \( 0.1(5.5) \):
\[ 0.55 + 0.3y = 0.9 \]
Now, solve for \( 0.3y \):
\[ 0.3y = 0.9 - 0.55 \]
\[ 0.3y = 0.35 \]
Step 4: Solve for \( y \):
\[ y = \frac{0.35}{0.3} = \frac{35}{30} = \frac{7}{6} \]
Now we have the solution:
\[ (x, y) = (5.5, \frac{7}{6}) \]
In fractional form, \( 5.5 = \frac{11}{2} \).
Therefore, the solution to the system of equations is:
\[ \left( \frac{11}{2}, \frac{7}{6} \right) \]
The correct response option is:
\(\left( \frac{11}{2}, \frac{7}{6} \right)\)