To solve the system of equations using substitution, we first rewrite one of the equations in terms of one variable and then substitute it into the other equation. Let's go through the steps:
Given equations:
- \( \frac{1}{3}x + y = 3 \)
- \( 2y = -\frac{2}{3}x + 10 \)
Step 1: Solve the first equation for \(y\).
From the first equation: \[ y = 3 - \frac{1}{3}x \]
Step 2: Substitute \(y\) into the second equation.
Now, we substitute \(y\) into the second equation: \[ 2(3 - \frac{1}{3}x) = -\frac{2}{3}x + 10 \]
Step 3: Simplify and solve for \(x\).
Distributing on the left side: \[ 6 - \frac{2}{3}x = -\frac{2}{3}x + 10 \]
We can add \(\frac{2}{3}x\) to both sides: \[ 6 = 10 \]
Step 4: Analyze the result.
Since the left side (6) does not equal the right side (10), this indicates that there is no solution to the system of equations. The lines represented by these equations are parallel and do not intersect.
Conclusion:
The system of equations has no solution, meaning the two lines do not intersect.