To solve the system of equations using the elimination method, we first need to rewrite the equations in standard form (Ax + By = C) and then manipulate them to eliminate one variable.
The original equations are:
- \( \frac{1}{3}x + y = 3 \)
- \( 2y = -\frac{2}{3}x + 10 \)
Step 1: Rewrite both equations in standard form
Equation 1: Multiply every term by 3 to eliminate the fraction:
\[ 3\left(\frac{1}{3}x\right) + 3y = 3 \times 3 \] \[ x + 3y = 9 \tag{1} \]
Equation 2: First, rewrite it in a standard form by moving all terms to one side. We can do this by adding \( \frac{2}{3}x \) to both sides:
\[ 2y + \frac{2}{3}x = 10 \]
Now, multiply every term by 3 to eliminate the fraction:
\[ 3(2y) + 3\left(\frac{2}{3}x\right) = 3 \times 10 \] \[ 6y + 2x = 30 \]
Rearranging gives:
\[ 2x + 6y = 30 \tag{2} \]
Step 2: Prepare for elimination
Now we have:
- \( x + 3y = 9 \)
- \( 2x + 6y = 30 \)
Step 3: Use substitution or elimination
We will multiply Equation (1) by 2 to align the coefficients of \(x\) for elimination.
\[ 2(x + 3y) = 2 \times 9 \]
This gives us the new equation:
\[ 2x + 6y = 18 \tag{3} \]
Step 4: Subtract the equations
Now we have:
- Equation (2): \( 2x + 6y = 30 \)
- Equation (3): \( 2x + 6y = 18 \)
Subtract Equation (3) from Equation (2):
\[ (2x + 6y) - (2x + 6y) = 30 - 18 \] \[ 0 = 12 \]
Conclusion
The result \( 0 = 12 \) is a contradiction. This means that the original system of equations has no solution. Therefore, the two lines represented by the equations are parallel and do not intersect.
Final Result
The system of equations is inconsistent, indicating that there are no solutions.