We are given the following system of equations:

$a^2 + b^2 + c^2 = 25$ $x^2 + y^2 + z^2 = 36$ $ax + by + cz = 30$ $\frac{a + b + x}{x + y + z} = m$

Step 1: Let’s simplify the expression for $m$

We need to find $m$, where $m = \frac{a + b + x}{x + y + z}$.

From the second equation, we know that $x^2 + y^2 + z^2 = 36$. This implies that: $x + y + z = \sqrt{36} = 6$

Now the expression for $m$ becomes: $m = \frac{a + b + x}{6}$

Step 2: Solve for $a + b + x$

We need to calculate $a + b + x$ to find $m$. Using the information from the first three equations, we aim to find values or relationships between $a$, $b$, $c$, $x$, $y$, and $z$.

Step 3: Geometrical Interpretation and Application of the Cauchy-Schwarz Inequality

The equations resemble the form of vectors, and applying the Cauchy-Schwarz inequality will help us determine the maximum or minimum values.

For the vectors $\mathbf{v} = (a, b, c)$ and $\mathbf{u} = (x, y, z)$, the Cauchy-Schwarz inequality states: $(ax + by + cz)^2 \leq (a^2 + b^2 + c^2)(x^2 + y^2 + z^2)$

Substitute the known values: $30^2 \leq 25 \times 36$ $900 \leq 900$ This shows that equality holds, meaning that the vectors $\mathbf{v}$ and $\mathbf{u}$ are collinear. Therefore, we can express one vector as a scalar multiple of the other: $(a, b, c) = k(x, y, z)$ for some constant $k$.

Step 4: Determine $k$

Using $ax + by + cz = 30$, we substitute $a = kx$, $b = ky$, and $c = kz$: $kx^2 + ky^2 + kz^2 = 30$ $k(x^2 + y^2 + z^2) = 30$ $k \times 36 = 30 \quad \Rightarrow \quad k = \frac{30}{36} = \frac{5}{6}$

Step 5: Solve for $a$, $b$, and $c$

Now, using $a = \frac{5}{6}x$, $b = \frac{5}{6}y$, and $c = \frac{5}{6}z$, substitute into the equation $a^2 + b^2 + c^2 = 25$: $\left( \frac{5}{6}x \right)^2 + \left( \frac{5}{6}y \right)^2 + \left( \frac{5}{6}z \right)^2 = 25$ $\frac{25}{36}(x^2 + y^2 + z^2) = 25$ $\frac{25}{36} \times 36 = 25$ This confirms the values of $a$, $b$, and $c$.

Step 6: Find $m$

We can now calculate $a + b + x$: $a + b + x = \frac{5}{6}x + \frac{5}{6}y + x = \frac{11}{6}x + \frac{5}{6}y$ Thus, using all terms computed