a^2+b^2+c^2=25, x^2+y^2+z^2=36, ax+by+cz=30, (a+b+x)/(x+y+z)=m. Find m

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

a^2 + b^2 + c^2 = 25, x^2 + y^2 + z^2 = 36, ax + by + cz = 30, (a+b+x)/(x+y+z) = m. Find m.

Learn how to solve a system of equations involving vectors and the Cauchy-Schwarz inequality. This problem explores algebraic concepts and vector geometry to find the value of m. Suitable for high school students in Grades 10-12.

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