If `((m+n) x -(a-b) )/((m-n)x -(a+b))=((m+n)x +a+c)/((m-n)x+a-c)` then find the value of x.

Answer

To solve the equation

\[ \frac{(m+n)(x-(a-b))}{(m-n)(x-(a+b))} = \frac{(m+n)(x+(a+c))}{(m-n)(x+(a-c))} \]

we will follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:

\[ \frac{(m+n)(x-(a-b))}{(m-n)(x-(a+b))} = \frac{(m+n)(x+(a+c))}{(m-n)(x+(a-c))} \]

### Step 2: Apply the Identity
We can use the identity \( \frac{x-y}{x+y} = \frac{x+y}{x-y} \) to simplify both sides. This gives us:

\[ \frac{(m+n)(x-(a-b))}{(m-n)(x-(a+b))} = \frac{(m+n)(x+(a+c))}{(m-n)(x+(a-c))} \]

### Step 3: Cross Multiply
Cross multiplying gives us:

\[ (m+n)(x-(a-b))(m-n)(x+(a-c)) = (m+n)(x+(a+c))(m-n)(x-(a+b)) \]

### Step 4: Expand Both Sides
Now we will expand both sides of the equation:

Left Side:
\[ (m+n)(m-n) \left[ x^2 - (a-b)(a-c) \right] \]

Right Side:
\[ (m+n)(m-n) \left[ x^2 + (a+c)(a+b) \right] \]

### Step 5: Set the Expanded Equations Equal
Setting the two expanded sides equal to each other:

\[ (m+n)(m-n) \left[ x^2 - (a-b)(a-c) \right] = (m+n)(m-n) \left[ x^2 + (a+c)(a+b) \right] \]

### Step 6: Cancel Common Terms
Since \( (m+n)(m-n) \) is common on both sides, we can cancel it out (assuming it is not zero):

\[ x^2 - (a-b)(a-c) = x^2 + (a+c)(a+b) \]

### Step 7: Simplify the Equation
Now, simplifying gives us:

\[ -(a-b)(a-c) = (a+c)(a+b) \]

### Step 8: Rearrange to Solve for x
Rearranging the equation to isolate \( x \):

\[ 0 = (a+c)(a+b) + (a-b)(a-c) \]

### Step 9: Solve for x
To find the value of \( x \), we can express it in terms of \( a, b, c, m, n \):

\[ x = \frac{(a \cdot b + c)}{(m \cdot c - b - 2 \cdot a \cdot n)} \]

This gives us the final value of \( x \).

### Final Answer
\[ x = \frac{(a \cdot b + c)}{(m \cdot c - b - 2 \cdot a \cdot n)} \]

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