To solve the equation
[
\frac{x - 3}{x} = \frac{2x - 1}{x} + 24
]
we can start by simplifying the equation. First, we can eliminate the fractions by multiplying both sides by ( x ) (assuming ( x \neq 0 )):
[
x(x - 3) = x(2x - 1) + 24x
]
This simplifies to:
[
x^2 - 3x = 2x^2 - x + 24x
]
Now, let's combine like terms on the right side:
[
x^2 - 3x = 2x^2 + 23x
]
Next, we move all terms to one side of the equation:
[
x^2 - 3x - 2x^2 - 23x = 0
]
This simplifies to:
[
-x^2 - 26x = 0
]
Factoring out a negative sign gives us:
[
-x(x + 26) = 0
]
Setting each factor equal to zero results in:
- ( -x = 0 ) → ( x = 0 ) (which we need to discard since this will make the original fractions undefined).
- ( x + 26 = 0 ) → ( x = -26 )
So, the solution to the equation is:
[
x = -26
]
Finally, double-checking the original equation to ensure that ( x = -26 ) does not lead to any undefined terms (like division by zero), we can conclude that ( x = -26 ) is valid and the solution:
[
\boxed{-26}
]
