To solve the system using the elimination method, let's first rewrite the equations based on your input:
- \(\frac{1}{5} x + y = \frac{6}{5}\)
- \(\frac{1}{10} x + \frac{1}{3} y = \frac{7}{10}\)
Next, we'll eliminate the fractions by multiplying both equations by the least common multiple (LCM) of the denominators.
Step 1: Eliminate Fractions
For the first equation, the LCM of 5 is 5. Multiply the entire equation by 5:
\[ 5\left(\frac{1}{5} x + y\right) = 5\left(\frac{6}{5}\right) \]
This simplifies to:
\[ x + 5y = 6 \quad \text{(Equation 1)} \]
For the second equation, the LCM of 10 and 3 is 30. Multiply the entire equation by 30:
\[ 30\left(\frac{1}{10} x + \frac{1}{3} y\right) = 30\left(\frac{7}{10}\right) \]
This simplifies to:
\[ 3x + 10y = 21 \quad \text{(Equation 2)} \]
Step 2: Set Up the New System
Now we have the following system of equations:
- \(x + 5y = 6\)
- \(3x + 10y = 21\)
Step 3: Elimination Method
We can multiply Equation 1 by 3 to eliminate \(x\):
\[ 3(x + 5y) = 3(6) \] which gives: \[ 3x + 15y = 18 \quad \text{(Equation 3)} \]
Now we have the modified system:
- \(3x + 15y = 18\) (from Equation 1)
- \(3x + 10y = 21\) (Equation 2)
Step 4: Subtract Equations
Subtract Equation 2 from Equation 3:
\[ (3x + 15y) - (3x + 10y) = 18 - 21 \]
This simplifies to:
\[ 5y = -3 \]
Step 5: Solve for \(y\)
Divide both sides by 5:
\[ y = -\frac{3}{5} \]
Step 6: Substitute \(y\) Back to Find \(x\)
Now substitute \(y\) back into one of the original equations. We'll use Equation 1:
\[ x + 5\left(-\frac{3}{5}\right) = 6 \]
This simplifies to:
\[ x - 3 = 6 \]
Adding 3 to both sides gives:
\[ x = 9 \]
Step 7: Final Solution
The solution to the system is:
\[ x = 9, \quad y = -\frac{3}{5} \]
Thus, the final answer is:
\[ (x, y) = \left(9, -\frac{3}{5}\right) \]