1. **State the problem:** We are given two expressions for angles: $(5x + 23)^\circ$ and $(17x - 41)^\circ$, and we want to find the value of $x$ when these two angles are equal. 2. **Set up the equation:** Since the angles are equal, we write: $$5x + 23 = 17x - 41$$ 3. **Isolate the variable $x$:** Subtract $5x$ from both sides: $$\cancel{5x} + 23 = 17x - \cancel{5x} - 41$$ which simplifies to $$23 = 12x - 41$$ 4. **Add 41 to both sides:** $$23 + 41 = 12x - 41 + 41$$ $$64 = 12x$$ 5. **Divide both sides by 12:** $$\frac{64}{\cancel{12}} = x \frac{\cancel{12}}{12}$$ which simplifies to $$x = \frac{64}{12} = \frac{16}{3} \approx 5.33$$ **Final answer:** $$x = \frac{16}{3}$$