1. **State the problem:** Given two parallel lines $m \parallel n$ cut by a transversal $t$, find the value of $x$ given the angles $(5x+23)^\circ$ and $(6x-4)^\circ$. 2. **Identify the relationship between the angles:** Since $m$ and $n$ are parallel and $t$ is a transversal, the angles given are alternate interior angles, which are congruent. 3. **Set up the equation:** $$5x + 23 = 6x - 4$$ 4. **Solve for $x$:** Subtract $5x$ from both sides: $$\cancel{5x} + 23 = \cancel{5x} + 6x - 4 \implies 23 = x - 4$$ Add 4 to both sides: $$23 + 4 = x - 4 + 4 \implies 27 = x$$ 5. **Final answer:** $$\boxed{27}$$