**For the System of Linear Equations:**
1. Start with the given system of equations.
2. Create the augmented matrix with coefficients and the right-hand side.
3. Perform row operations to transform the matrix into echelon form.
4. Continue with row operations to simplify the matrix further.
5. Solve for the variables using back-substitution.
**For the Movie Theater Problem:**
1. Define the variables: a for adults, c for children, s for students.
2. Set up equations based on the problem’s conditions.
3. Write the equations in a standard form with all variables on one side.
4. Create the augmented matrix with coefficients and the right-hand side.
5. Perform row operations to convert the matrix into echelon form.
6. Continue with row operations to simplify the matrix further.
7. Solve for the variables using back-substitution.
Your solution method avoids substitution until the final step, where you have a matrix in the desired form. Make sure to include this form in your notes as requested. Your homework solution should contain between 15 and 20 steps, and you should demonstrate your understanding of the row operations used to solve the system.