1. **State the problem:** We have a rectangle ABCD with sides labeled as follows: - AB = $2x + 5y$ - BC = $x + 1$ - AD = $3y$ - DC = $x + 4y + 7$ We need to: a) Write two equations relating $x$ and $y$. b) Solve these equations to find $x$ and $y$. c) Find the area of the rectangle. 2. **Form the equations:** Since ABCD is a rectangle, opposite sides are equal in length. - So, AB = DC gives: $$2x + 5y = x + 4y + 7$$ - And AD = BC gives: $$3y = x + 1$$ 3. **Simplify the first equation:** $$2x + 5y = x + 4y + 7$$ Subtract $x$ and $4y$ from both sides: $$2x - \cancel{x} + 5y - \cancel{4y} = \cancel{x} + 4y - \cancel{4y} + 7$$ $$x + y = 7$$ 4. **Rewrite the second equation:** $$3y = x + 1$$ Rearranged: $$x - 3y = -1$$ 5. **Solve the system using elimination:** Equations: $$x + y = 7$$ $$x - 3y = -1$$ Subtract the second from the first to eliminate $x$: $$\cancel{x} + y - (\cancel{x} - 3y) = 7 - (-1)$$ $$y - (-3y) = 7 + 1$$ $$y + 3y = 8$$ $$4y = 8$$ $$y = 2$$ 6. **Find $x$ using $x + y = 7$:** $$x + 2 = 7$$ $$x = 5$$ 7. **Calculate the area of the rectangle:** Area = length × width Using AB and BC: $$\text{Area} = (2x + 5y)(x + 1)$$ Substitute $x=5$, $y=2$: $$= (2(5) + 5(2))(5 + 1)$$ $$= (10 + 10)(6)$$ $$= 20 \times 6 = 120$$ **Final answers:** - $x = 5$ - $y = 2$ - Area = 120 cm²