School plot modification

When modifying the school plot in the shape of a rectangle, the deviation was calculated if we increased the length and width of the plot by 1 m and its area by 22 m². If we reduce the length of the plot by 2 m and increase its width by 1 m, its area will decrease by 5 m². What is the size of the school grounds?

Final Answer:

S =  104 m²

Step-by-step explanation:

$$\begin{gathered} S=ab \\ (a+1)\cdot (b+1)=S+22 \\ (a-2)\cdot (b+1)=S-5 \\ {a}^2+2ab+b^2=ab+22 \\ ab+a-2b-2=ab-5 \\ ab+a-2b-2=ab-5 \\ a-2b-2=-5 \\ a=2b-3 \\ (2b-3+1)\cdot (b+1)=(2b-3)\cdot b+22 \\ 2b^2-2=2b^2-3b+22 \\ -2=-3b+22 \\ -2=-3\cdot b+22 \\ 3b=24 \\ b=\frac{24}{3}=8 \\ b=8 \\ a=2\cdot b-3=2\cdot 8-3=13\text{m} \\ S=a\cdot b=13\cdot 8=104 \\ \text{ Verifying Solution: } \\ {S}_1=(a+1)\cdot (b+1)=(13+1)\cdot (8+1)=126{\text{m}}^2 \\ {S}_2=(a-2)\cdot (b+1)=(13-2)\cdot (8+1)=99{\text{m}}^2 \end{gathered}$$

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