$$ \frac { a } { a x - 1 } + \frac { b } { b u - 1 } - a + b = 0 $$

Question

UpStudy AI · Accepted Answer

Certainly! Let's solve the equation for $$ x $$:

$$
\frac{a}{a x - 1} + \frac{b}{b u - 1} - a + b = 0
$$

### Step 1: Rearrange the Equation

First, move the constants to the other side:

$$
\frac{a}{a x - 1} + \frac{b}{b u - 1} = a - b
$$

### Step 2: Combine the Fractions

To combine the fractions, find a common denominator, which is $$ (a x - 1)(b u - 1) $$:

$$
\frac{a(b u - 1) + b(a x - 1)}{(a x - 1)(b u - 1)} = a - b
$$

### Step 3: Cross-Multiply

Multiply both sides by the denominator to eliminate the fraction:

$$
a(b u - 1) + b(a x - 1) = (a - b)(a x - 1)(b u - 1)
$$

### Step 4: Expand Both Sides

Let's expand both sides of the equation.

**Left Side:**

$$
a b u - a + a b x - b
$$

**Right Side:**

$$
(a - b)(a x - 1)(b u - 1) = (a - b)(a x b u - a x - b u + 1)
$$

Expanding further:

$$
= a(a x b u - a x - b u + 1) - b(a x b u - a x - b u + 1)
$$
$$
= a^2 b u x - a^2 x - a b u + a - a b^2 u x + a b x + b^2 u - b
$$

### Step 5: Bring All Terms to One Side

Set the equation to zero by moving all terms to the left side:

$$
a b u - a + a b x - b - (a^2 b u x - a^2 x - a b u + a - a b^2 u x + a b x + b^2 u - b) = 0
$$

Simplify the equation:

$$
2 a b u - 2 a - b^2 u + a^2 x - a b u x + a b^2 u x = 0
$$

### Step 6: Solve for $$ x $$

Factor out $$ x $$ from the terms involving it:

$$
x(a^2 + a b^2 u - a b u) = 2 a - 2 a b u + b^2 u
$$

Finally, solve for $$ x $$:

$$
x = \frac{2 a - 2 a b u + b^2 u}{a^2 + a b^2 u - a b u}
$$

### Step 7: Simplify the Expression

Factor where possible to simplify:

$$
x = \frac{2a(1 - b u) + b^2 u}{a(a + b u (b - 1))}
$$

Or, alternatively:

$$
x = \frac{2a + u b(b - 2a)}{a^2 + a b u (b - 1)}
$$

### Final Answer

$$
x = \frac{2a + u b (b - 2a)}{a^2 + a b u (b - 1)}
$$

This expression gives $$ x $$ in terms of $$ a $$, $$ b $$, and $$ u $$.
Solve for $$ x $$: $$ x = \frac{2a + u b (b - 2a)}{a^2 + a b u (b - 1)} $$

\( \frac { a } { a x - 1 } + \frac { b } { b u - 1 } - a + b = 0 \)