For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.
2. 
1/2x−1/3y=4 3/2x−y=0

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.

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists. 2. 1/2x−1/3y=4 3/2x−y=0

Learn how to solve a system of linear equations using substitution or elimination methods. This example demonstrates how to handle a system where no solution exists due to a contradiction. Suitable for students in Grades 8-10.

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