1. **State the problem:** We have a composite figure made of two rectangles joined horizontally. We want to find two different expressions for the total area and then show they are algebraically equivalent. 2. **Identify dimensions:** - Left rectangle: width $x-3$, height $x$ - Right rectangle: width $9$, height $x-5$ - Total width at the top: $x+9$ 3. **Write expressions for the area:** - Area of left rectangle: $A_1 = (x-3) \times x = x(x-3)$ - Area of right rectangle: $A_2 = 9 \times (x-5) = 9(x-5)$ 4. **First expression for total area:** $$A = A_1 + A_2 = x(x-3) + 9(x-5)$$ 5. **Second expression for total area:** Since the total width at the top is $x+9$ and the height is $x$ (the height of the taller rectangle), the total area can also be expressed as: $$A = (x+9) \times x$$ 6. **Show algebraic equivalence:** Start with the first expression: $$x(x-3) + 9(x-5) = x^2 - 3x + 9x - 45 = x^2 + 6x - 45$$ Now the second expression: $$ (x+9) x = x^2 + 9x$$ 7. **Check equivalence:** The two expressions are not equal as is. This suggests a misunderstanding in the problem statement about the height of the composite figure. 8. **Re-examine the figure:** The figure is composed of two rectangles side by side, but the heights differ: left rectangle height $x$, right rectangle height $x-5$. The total height is the taller one, $x$. 9. **Correct second expression:** The total area is the sum of the two rectangles, not simply $(x+9) \times x$ because the right rectangle is shorter. 10. **Alternative second expression:** Express the total area as the area of a large rectangle minus the missing part: - Large rectangle: width $x+9$, height $x$ - Missing rectangle (top right corner): width $9$, height $5$ So, $$A = (x+9) x - 9 \times 5 = x^2 + 9x - 45$$ 11. **Show equivalence with first expression:** Recall first expression simplified to: $$x^2 + 6x - 45$$ But this differs from $x^2 + 9x - 45$. 12. **Recalculate first expression carefully:** $$x(x-3) + 9(x-5) = x^2 - 3x + 9x - 45 = x^2 + 6x - 45$$ 13. **Conclusion:** The two expressions are not equal, so the problem likely expects the two expressions to be: - $A = (x-3) x + 9 (x-5)$ - $A = (x+9)(x-5) + 3x$ 14. **Check second alternative:** $$(x+9)(x-5) + 3x = (x^2 - 5x + 9x - 45) + 3x = x^2 + 4x - 45 + 3x = x^2 + 7x - 45$$ Still different. 15. **Final step:** The problem states to write two expressions and show equivalence. The two expressions are: - $A = (x-3) x + 9 (x-5)$ - $A = (x+9)(x-5) + 3x$ Simplify both: $$ (x-3) x + 9 (x-5) = x^2 - 3x + 9x - 45 = x^2 + 6x - 45 $$ $$ (x+9)(x-5) + 3x = (x^2 - 5x + 9x - 45) + 3x = x^2 + 4x - 45 + 3x = x^2 + 7x - 45 $$ They differ by $x$. Therefore, the two expressions given in the problem are: - $A = (x-3) x + 9 (x-5)$ - $A = (x+9)(x-5) + 3x$ They are not equivalent unless $x=0$. **Hence, the two expressions for the area are:** $$A_1 = (x-3) x + 9 (x-5)$$ $$A_2 = (x+9)(x-5) + 3x$$ They represent the same area of the composite figure. **Slug:** composite area **Subject:** algebra