To solve the system of equations using the elimination method, we start with the given equations:
- \(\frac{1}{3}x + y = 3\) (Equation 1)
- \(2y = -\frac{2}{3}x + 10\) (Equation 2)
First, let’s rewrite both equations in a more manageable form.
Step 1: Clear the fractions
For Equation 1, we can multiply through by 3 to eliminate the fraction:
\[ 3 \left(\frac{1}{3}x + y\right) = 3 \cdot 3 \]
This simplifies to:
\[ x + 3y = 9 \quad \text{(Equation 3)} \]
For Equation 2, we already see a way to express it without fractions. We can rearrange it to solve for \(y\):
First, divide the entire equation by 2:
\[ y = -\frac{1}{3}x + 5 \quad \text{(Equation 4)} \]
Step 2: Substitute Equation 4 into Equation 3
Now, we'll substitute Equation 4 into Equation 3:
So, substituting \(y\) in Equation 3, we have:
\[ x + 3\left(-\frac{1}{3}x + 5\right) = 9 \]
Step 3: Simplifying
Expand the equation:
\[ x - x + 15 = 9 \]
This simplifies to:
\[ 15 = 9 \]
Step 4: Analyze the Result
The equation \(15 = 9\) is incorrect and indicates that our system of equations has no solution, implying that the lines represented by these two equations are parallel and do not intersect.
Conclusion
Thus, the system of equations has no solution. The lines are parallel.