Let's address the two questions:

Question 27: Simplify the circuit by writing its logical expression and conclusion

The circuit consists of switches $S_1$, $S_2$, and $S_3$. It appears to be a combination of series and parallel connections.

1. Logical Behavior:

   • When switches are in series, their operation corresponds to a logical AND gate.
   • When switches are in parallel, their operation corresponds to a logical OR gate.

   Based on the diagram:

   • $S_1$ and $S_2$ are in parallel, so their operation can be written as $S_1 + S_2$ (OR logic).
   • $S_3$ is in series with the parallel combination, so the total logical expression becomes: $(S_1 + S_2) \cdot S_3$ (AND logic between $S_3$ and $S_1 + S_2$).

2. Conclusion:

   • The circuit operates (output is ON) if either $S_1$ or $S_2$ is closed AND $S_3$ is closed.

Question 28: Solve the system of equations by the method of reduction

The system of equations is:

1. $x - y + z = 1$
2. $2x - y = 1$
3. $3x + 3y - 4z = 2$

Step 1: Simplify the equations

From equation (2): $2x - y = 1 \quad \implies \quad y = 2x - 1$

Substitute $y = 2x - 1$ into equations (1) and (3).

Step 2: Substitute into (1)

$x - (2x - 1) + z = 1$ $x - 2x + 1 + z = 1 \quad \implies \quad -x + z = 0 \quad \implies \quad z = x$

Step 3: Substitute $y = 2x - 1$ and $z = x$ into (3)

$3x + 3(2x - 1) - 4(x) = 2$ $3x + 6x - 3 - 4x = 2$ $5x - 3 = 2 \quad \implies \quad 5x = 5 \quad \implies \quad x = 1$

Step 4: Solve for $y$ and $z$

• From $y = 2x - 1$: $y = 2(1) - 1 = 1$
• From $z = x$: $z = 1$

Final Solution:

$x = 1, \, y = 1, \, z = 1$

Would you like further explanations or verification? Here are 5 related questions for further exploration:

1. Can you analyze a more complex logical circuit with additional switches?
2. How would the solution change in the system of equations if one equation was altered?
3. Can you write the truth table for the given circuit in Question 27?
4. How does the method of reduction differ from substitution in solving systems of equations?
5. What are the practical applications of circuits modeled by logical expressions?

Tip: For logical circuits, always identify the series and parallel sections first to simplify their expressions systematically.