1. **State the problem:** We have multiple lawn sections with given algebraic dimensions and know their areas are equal. We need to find the dimensions of each section. 2. **Identify the shapes and their areas:** - Triangle with base $3x - 7$ and height $2x - 4$. Area formula: $$\text{Area} = \frac{b \times h}{2}$$ - Rectangle with sides $4xw$ and $x - 1$. Area formula: $$\text{Area} = \text{length} \times \text{width}$$ - Another rectangle with sides $(y - 1)$ and $(2x - 9)$. Area formula: $$\text{Area} = (y - 1)(2x - 9)$$ - Rectangle with side $3x - 9$ (only one side given, so assume square or missing info; we will use given equalities). 3. **Write expressions for each area:** - Triangle area: $$A_1 = \frac{(3x - 7)(2x - 4)}{2}$$ - Rectangle 1 area: $$A_2 = 4xw (x - 1)$$ - Rectangle 2 area: $$A_3 = (y - 1)(2x - 9)$$ - Rectangle 3 area: $$A_4 = (3x - 9) \times \text{(missing side)}$$ (Assuming equal to others, we focus on given equalities.) 4. **Set areas equal:** Since the problem states all areas are equal, set $A_1 = A_2 = A_3$. 5. **Simplify $A_1$:** $$A_1 = \frac{(3x - 7)(2x - 4)}{2} = \frac{6x^2 - 12x - 14x + 28}{2} = \frac{6x^2 - 26x + 28}{2} = 3x^2 - 13x + 14$$ 6. **Express $A_2$:** $$A_2 = 4xw (x - 1) = 4xw x - 4xw = 4x^2 w - 4x w$$ 7. **Express $A_3$:** $$A_3 = (y - 1)(2x - 9) = 2xy - 9y - 2x + 9$$ 8. **Set $A_1 = A_2$:** $$3x^2 - 13x + 14 = 4x^2 w - 4x w$$ 9. **Set $A_1 = A_3$:** $$3x^2 - 13x + 14 = 2xy - 9y - 2x + 9$$ 10. **Solve for $w$ from $A_1 = A_2$:** Group terms: $$4x^2 w - 4x w = 3x^2 - 13x + 14$$ Factor $w$: $$w(4x^2 - 4x) = 3x^2 - 13x + 14$$ $$w = \frac{3x^2 - 13x + 14}{4x^2 - 4x}$$ 11. **Solve for $y$ from $A_1 = A_3$:** Rearranged: $$2xy - 9y = 3x^2 - 13x + 14 + 2x - 9$$ $$y(2x - 9) = 3x^2 - 11x + 5$$ $$y = \frac{3x^2 - 11x + 5}{2x - 9}$$ 12. **Summary:** - Base and height of triangle: $3x - 7$, $2x - 4$ - Rectangle 1 sides: $4x w$, $x - 1$ with $$w = \frac{3x^2 - 13x + 14}{4x^2 - 4x}$$ - Rectangle 2 sides: $y - 1$, $2x - 9$ with $$y = \frac{3x^2 - 11x + 5}{2x - 9}$$ These expressions give the dimensions of each section in terms of $x$. **Final answer:** $$\boxed{\text{Dimensions depend on } x:\quad w = \frac{3x^2 - 13x + 14}{4x^2 - 4x}, \quad y = \frac{3x^2 - 11x + 5}{2x - 9}}$$