If ab+ac+bc =1 prove that 2/abc + 9abc = 7(a+b+c)

We are given the equation:

\[
ab + ac + bc = 1
\]

We need to prove the following:

\[
\frac{2}{abc} + 9abc = 7(a + b + c)
\]

### Step 1: Assume variables
Let's denote:

\[
S = a + b + c, \quad P = abc, \quad Q = ab + ac + bc = 1
\]

This simplifies the given information.

### Step 2: Express the terms
We aim to prove the equation:

\[
\frac{2}{P} + 9P = 7S
\]

Substituting \(P\) and \(S\) for the respective terms simplifies our expression. Next, we observe that the left-hand side involves both \(\frac{1}{P}\) and \(P\).

### Step 3: Check for consistency
Now, we need more information to prove this directly, as it appears we would need to solve or further manipulate \(a\), \(b\), and \(c\). Without further information or constraints on \(a\), \(b\), and \(c\), this might indicate that more specific values or additional constraints on the variables are necessary.

Would you like to proceed by exploring particular values or further decomposing the problem?

### Questions to explore:
1. What are possible values for \(a\), \(b\), and \(c\)?
2. Can you derive other identities using symmetry in \(a\), \(b\), and \(c\)?
3. How does the equation change with specific assumptions about \(a = b = c\)?
4. What happens when we try factoring terms from both sides?
5. Is there a geometric interpretation for this system?

### Tip: 
When working with systems involving multiple variables, symmetries and special cases often reveal useful insights.

If ab + ac + bc = 1, prove that 2/abc + 9abc = 7(a + b + c).

Explore the proof for the equation 2/abc + 9abc = 7(a + b + c), given that ab + ac + bc = 1. This problem involves algebraic symmetry, systems of equations, and factoring techniques, making it ideal for high school students (Grades 10-12).

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