x-3/x=2x-1/x+24
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:
\[ \boxed{-26} \]
