Let's solve the system of linear equations using substitution or elimination. The given system is:

\frac{1}{2}x - \frac{1}{3}y = 4 \tag{1} \frac{3}{2}x - y = 0 \tag{2}

Step 1: Solve Equation (2) for $y$

From Equation (2): $\frac{3}{2}x - y = 0$ Solve for $y$: y = \frac{3}{2}x \tag{3}

Step 2: Substitute Equation (3) into Equation (1)

Now, substitute $y = \frac{3}{2}x$ from Equation (3) into Equation (1): $\frac{1}{2}x - \frac{1}{3} \left(\frac{3}{2}x\right) = 4$

Simplify: $\frac{1}{2}x - \frac{1}{3} \times \frac{3}{2}x = 4$ $\frac{1}{2}x - \frac{1}{2}x = 4$ $0 = 4$

Step 3: Conclusion

Since we arrived at a contradiction ($0 = 4$), the system of equations has no solution.

Final Answer:

The system has no solution.

Would you like further clarification or help with another problem?

Related Questions:

1. What is the geometric interpretation of a system of equations with no solution?
2. How do you recognize when a system of equations has infinitely many solutions?
3. Can you solve a system of linear equations using matrices?
4. What is the difference between substitution and elimination methods?
5. How do you solve a system of nonlinear equations?

Tip:

When solving a system of equations, always check for contradictions or identity situations (like $0 = 0$) that indicate special cases like no solution or infinitely many solutions.