The problem in the image provides the following system of equations involving real numbers $a$, $b$, and $c$:

$a^2 + 27 = -8b + 10c$ $b^2 + 196 = 18c + 13a$ $c^2 + 119 = -3a + 30b$

We are asked to find the value of $a + 3b + 5c$.

Steps to Solve:

1. First, we will simplify and rearrange each equation to get expressions for $a^2$, $b^2$, and $c^2$ in terms of $a$, $b$, and $c$.
2. Then, using substitution or elimination, we will attempt to solve the system step by step.
3. Finally, we will substitute the values of $a$, $b$, and $c$ into the expression $a + 3b + 5c$ to find the required value.

Let me begin solving it systematically.

The system of equations yields the solution $a = 5$, $b = 11$, and $c = 14$.

Substituting these values into the expression $a + 3b + 5c$, we get:

$a + 3b + 5c = 5 + 3(11) + 5(14) = 5 + 33 + 70 = 108$

Thus, the value of $a + 3b + 5c$ is $108$.

Would you like more details or have any other questions?

Here are five related questions:

1. How do you solve systems of nonlinear equations?
2. What are the conditions under which systems of equations have unique solutions?
3. How would the solution change if we modified one of the coefficients in the equations?
4. Can a system of nonlinear equations have no solution?
5. What is the significance of solving such systems in real-world problems?

Tip: Always double-check each equation when solving a system, especially for sign errors or misplaced terms.