To simplify the expression \( \frac{ax - ay + bx - by}{ax - ay - bx + by} \), we can start by rearranging both the numerator and the denominator.
Step 1: Group terms in the numerator \[ ax - ay + bx - by = a(x - y) + b(x - y) = (a + b)(x - y) \]
Step 2: Group terms in the denominator \[ ax - ay - bx + by = a(x - y) - b(x - y) = (a - b)(x - y) \]
Step 3: Rewrite the expression Now we can rewrite the entire expression as: \[ \frac{(a + b)(x - y)}{(a - b)(x - y)} \]
Step 4: Cancel common factors If \( x - y \neq 0 \), we can cancel \( x - y \) from the numerator and denominator: \[ \frac{a + b}{a - b} \]
Thus, the simplified form of the expression \( \frac{ax - ay + bx - by}{ax - ay - bx + by} \) is: \[ \frac{a + b}{a - b}, \quad (x \neq y) \]