To solve the given system of linear equations and determine the number of solutions, let’s rewrite the system and analyze it:

The system of equations is:

1. $-3x - 5y + 36z = 10$
2. $-x + 7z = 5$
3. $x + y - 10z = -4$

Step 1: Analyze and reduce the system

We will solve this using substitution and elimination to see if the system has a unique solution, infinitely many solutions, or no solution.

Equation 2 ($-x + 7z = 5$) can be rewritten as:

$x = 7z - 5$

Substitute $x = 7z - 5$ into Equations 1 and 3:

Substituting into Equation 1: $-3(7z - 5) - 5y + 36z = 10$ $-21z + 15 - 5y + 36z = 10$ $15z - 5y + 15 = 10$ $15z - 5y = -5$ $3z - y = -1 \quad \text{(Equation 4)}$

Substituting into Equation 3: $(7z - 5) + y - 10z = -4$ $7z - 5 + y - 10z = -4$ $-3z + y = 1 \quad \text{(Equation 5)}$

Step 2: Solve the reduced system (Equations 4 and 5)

The reduced system is:

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

Add the two equations: $(3z - y) + (-3z + y) = -1 + 1$ $0 = 0$

This indicates that the two equations are dependent and represent the same line. Therefore, the system has infinitely many solutions along this line.

Step 3: Parametrize the solution

Let $z = t$ (a parameter), then:

1. From Equation 4: $y = 3z + 1 = 3t + 1$
2. From $x = 7z - 5$: $x = 7t - 5$

Thus, the solutions can be written as: $x = 7t - 5, \quad y = 3t + 1, \quad z = t$ where $t \in \mathbb{R}$.

Conclusion

The system has infinitely many solutions, parametrized as shown above.

Would you like more details about the solution, or have any specific questions? Here are related questions for further exploration:

1. How do we determine if a system of equations has no solution?
2. What are the key differences between dependent and independent equations in a system?
3. How can we verify the parametrized solution by substitution into the original equations?
4. What are some real-world scenarios where systems of equations arise?
5. How does the geometry of planes relate to the solution type for systems of three equations?

Tip: Always reduce the system of equations step-by-step to identify dependencies or inconsistencies!