1. **State the problem:** We are given a parallelogram with sides labeled as follows: top side = $12 - c$, left side = $2b$, bottom side = $c - 2$, right side = $4b - 6$. We need to find the perimeter of the parallelogram. 2. **Recall the formula for the perimeter of a parallelogram:** $$\text{Perimeter} = 2(\text{length} + \text{width})$$ In a parallelogram, opposite sides are equal in length. 3. **Identify equal sides:** - Top side = Bottom side, so $12 - c = c - 2$ - Left side = Right side, so $2b = 4b - 6$ 4. **Solve for $c$:** $$12 - c = c - 2$$ Add $c$ to both sides: $$12 = 2c - 2$$ Add 2 to both sides: $$12 + 2 = 2c$$ $$14 = 2c$$ Divide both sides by 2: $$\cancel{2}c / \cancel{2} = 14 / 2$$ $$c = 7$$ 5. **Solve for $b$:** $$2b = 4b - 6$$ Subtract $4b$ from both sides: $$2b - 4b = -6$$ $$-2b = -6$$ Divide both sides by $-2$: $$\cancel{-2}b / \cancel{-2} = -6 / -2$$ $$b = 3$$ 6. **Calculate the lengths of the sides:** - Top side = $12 - c = 12 - 7 = 5$ - Left side = $2b = 2 \times 3 = 6$ 7. **Calculate the perimeter:** $$\text{Perimeter} = 2(\text{length} + \text{width}) = 2(5 + 6) = 2 \times 11 = 22$$ **Final answer:** The perimeter is 22.