If `(p+q-x)/(r ) +(p+r-x)/(q)+(q+r-x)/(p)+(3x)/(p+q+r)=0`, then x=….

Answer

To solve the equation

\[ \frac{p+q-x}{r} + \frac{p+r-x}{q} + \frac{q+r-x}{p} + \frac{3x}{p+q+r} = 0, \]

we will first find a common denominator and simplify the equation step by step.

### Step 1: Find a common denominator
The common denominator for the first three fractions is \( pqr \) (the product of \( p, q, \) and \( r \)). The last term has a denominator of \( p+q+r \). Thus, we can rewrite the equation as:

\[ \frac{(p+q-x) \cdot pq + (p+r-x) \cdot pr + (q+r-x) \cdot qr + \frac{3x \cdot pqr}{p+q+r}}{pqr} = 0. \]

### Step 2: Multiply through by the common denominator
Multiplying both sides by \( pqr(p+q+r) \) gives us:

\[ (p+q-x) \cdot pq(p+q+r) + (p+r-x) \cdot pr(p+q+r) + (q+r-x) \cdot qr(p+q+r) + 3x \cdot pqr = 0. \]

### Step 3: Expand each term
Now we expand each term:

1. \( (p+q-x) \cdot pq(p+q+r) = pq(p+q)(p+q+r) - pqx(p+q+r) \)
2. \( (p+r-x) \cdot pr(p+q+r) = pr(p+r)(p+q+r) - prx(p+q+r) \)
3. \( (q+r-x) \cdot qr(p+q+r) = qr(q+r)(p+q+r) - qrx(p+q+r) \)

### Step 4: Combine like terms
Combining all the terms gives us a polynomial in \( x \). Collect all the \( x \) terms on one side and constant terms on the other side.

### Step 5: Rearranging the equation
Rearranging the equation will yield:

\[ (pq(p+q)(p+q+r) + pr(p+r)(p+q+r) + qr(q+r)(p+q+r)) = (pq + pr + qr + 3)pqr. \]

### Step 6: Solve for \( x \)
From the rearranged equation, we can isolate \( x \):

\[ x \left( -pq(p+q+r) - pr(p+q+r) - qr(p+q+r) + 3pqr \right) = \text{(constant terms)}. \]

Thus, we can solve for \( x \):

\[ x = \frac{\text{(constant terms)}}{-pq(p+q+r) - pr(p+q+r) - qr(p+q+r) + 3pqr}. \]

### Final Step: Simplify
Finally, simplify the expression for \( x \) to get the final result.