1. **State the problem:** We are given that \(\angle ABC = \angle ACB\) (the angles are equal). The lengths are given as \(AE = 3x + 4\), \(CE = y - 3\), \(BE = 2x + 1\), and \(DE = 2y + 5\). We need to find the value of \(x\). 2. **Analyze the given information:** Since \(\angle ABC = \angle ACB\), triangle \(ABC\) is isosceles with sides opposite these angles equal. This means \(AB = AC\). 3. **Relate sides to given segments:** We weren't given \(AB\) or \(AC\) in terms of \(x\) and \(y\), but we do have segments \(AE\), \(BE\), \(CE\), and \(DE\). Without further context, assume \(E\) is a point on some segments, but likely the problem expects to connect these to find relations. 4. **Given no direct relation, we use the equality of those sides to equate expressions: Since \(\angle ABC = \angle ACB\), \(AB = AC\) and if we assign \(AB = BE + EA = 2x + 1 + 3x + 4 = 5x + 5\) and \(AC = CE + EA = y - 3 + 3x + 4 = 3x + y + 1\). Then from \(AB = AC\), we get: $$5x + 5 = 3x + y + 1$$ 5. **Simplify to express \(y\) in terms of \(x\):** $$5x + 5 = 3x + y + 1$$ $$5x + 5 - 3x - 1 = y$$ $$2x + 4 = y$$ 6. **Use given lengths \(CE = y - 3\) and \(DE = 2y + 5\). Possibly a relation between \(CE\) and \(DE\) is expected; if the problem implies \(CE = DE\) (because \(E\) is the midpoint or some condition similar), then: $$y - 3 = 2y + 5$$ 7. **Solve this for \(y\):** $$y - 3 = 2y + 5$$ $$y - 3 - 2y - 5 = 0$$ $$-y - 8 = 0$$ $$y = -8$$ 8. **Substitute \(y = -8\) back to \(y = 2x + 4\):** $$-8 = 2x + 4$$ $$2x = -8 - 4 = -12$$ $$x = -6$$ 9. **Final answer:** The value of \(x\) is \(-6\).