(x-7)(x-8)%3D(x-3)(x-7)(x-9).jpeg "(x-3)(x-7)(x-8)=(x-3)(x-7)(x-9)")
Solving the Equation (x-3)(x-7)(x-8) = (x-3)(x-7)(x-9)
This equation presents a unique challenge because it involves a product of three factors on each side. Let's break down the solution step by step:
1. Recognizing the Common Factors
The first step is to notice that both sides of the equation share the factors (x-3) and (x-7). This allows us to simplify the equation significantly.
2. Dividing Both Sides
We can divide both sides of the equation by (x-3)(x-7) as long as these factors are not equal to zero. This gives us:
(x-8) = (x-9)
3. Solving for x
Now we have a simple linear equation. Subtracting x from both sides yields:
-8 = -9
This result is a contradiction, indicating that there is no solution for x that satisfies the original equation.
4. Understanding the Implication
The absence of a solution implies that the initial assumption that (x-3)(x-7) ≠ 0 was incorrect. In other words, either x = 3 or x = 7 must be true for the equation to hold. However, substituting either value into the original equation results in a product of zero on both sides, making the equation true but not providing a solution for x.
Conclusion
The equation (x-3)(x-7)(x-8) = (x-3)(x-7)(x-9) has no solutions for x. This arises from the fact that the common factors (x-3) and (x-7) lead to a contradiction after simplification. The only values of x that make the equation true are x = 3 and x = 7, but these values don't provide a solution in the context of solving for x.