1. The problem states the side lengths of an equilateral triangle as expressions: $$3w + 8$$, $$5w - 14$$, and $$2w + 19$$. 2. Since all sides of an equilateral triangle are equal, we set these expressions equal to each other: $$3w + 8 = 5w - 14 = 2w + 19$$ 3. First, equate the first two expressions: $$3w + 8 = 5w - 14$$ 4. Solve for $$w$$: $$3w + 8 = 5w - 14$$ $$8 + 14 = 5w - 3w$$ $$22 = 2w$$ $$w = \frac{22}{2} = 11$$ 5. Now, check if this $$w$$ satisfies the third side equal to the others: Calculate $$2w + 19$$: $$2(11) + 19 = 22 + 19 = 41$$ Calculate $$3w + 8$$: $$3(11) + 8 = 33 + 8 = 41$$ Calculate $$5w - 14$$: $$5(11) - 14 = 55 - 14 = 41$$ 6. Since all three expressions equal 41, $$w = 11$$ is the correct solution. 7. The side length of the equilateral triangle is therefore $$41$$ units.