a)

\(\left(x+4\right)\cdot\left(y-1\right)=13\)

\(\Rightarrow\left(x+4\right)\cdot\left(y-1\right)\in\text{ Ư(13) = }\left\{1;13;-1;-13\right\}\)

Ta có bảng sau:

Vậy, ta có cặp \(\left\{x;y\right\}\) thỏa mãn \(\left\{3;14\right\};\left\{9;2\right\};\left\{-5;-12\right\};\left\{-17;0\right\}\)

b)

\(xy-3x+y=20\)

\(\Rightarrow xy-3x+y=17+3\)

\(\Rightarrow xy-3x+y-3=17\)

\(\Rightarrow\left(xy+y\right)-\left(3x+3\right)=17\)

\(\Rightarrow y\left(x+1\right)-3\left(x+1\right)=17\)

\(\Rightarrow\left(y-3\right)\cdot\left(x+1\right)=17\)

\(\Rightarrow\left(y-3\right)\cdot\left(x+1\right)\in\text{ Ư(17) = }\left\{1;17;-1;-17\right\}\)

Ta có bảng sau:

Vậy, ta có cặp \(\left\{x;y\right\}\) thỏa mãn \(\left\{0;20\right\};\left\{4;16\right\};\left\{-14;-2\right\};\left\{-18;2\right\}\)

`1,`

`538 - x = 275`

`\Rightarrow x = 538 - 275`

`\Rightarrow x = 263`

Vậy, `x = 263`

`2,`

`45 - 9x = 18`

`\Rightarrow 9x = 45 - 18`

`\Rightarrow 9x = 27`

`\Rightarrow x = 27 \div 9`

`\Rightarrow x = 3`

Vậy, `x = 3`

`3,`

`(5x - 9) \div 3 = 12`

`\Rightarrow 5x - 9 = 12. 3`

`\Rightarrow 5x - 9 = 36`

`\Rightarrow 5x = 36 + 9`

`\Rightarrow 5x = 45`

`\Rightarrow x = 45 \div 5`

`\Rightarrow x = 9`

Vậy, `x = 9.`

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`@` `\text {Ans}`

`\downarrow`

\(9x^{10}-7x^9=0\)

`\Leftrightarrow x^9(9x-7)=0`

`\Leftrightarrow `\(\left[{}\begin{matrix}x^9=0\\9x-7=0\end{matrix}\right.\)

`\Leftrightarrow `\(\left[{}\begin{matrix}x=0\\9x=7\end{matrix}\right.\)

`\Leftrightarrow `\(\left[{}\begin{matrix}x=0\\x=\dfrac{7}{9}\end{matrix}\right.\)

Vậy, nghiệm của đa thức là `x \in {0; 7/9}.`

Đề có phải là:

\(\dfrac{x+1}{2024}+\dfrac{x+2}{2025}+\dfrac{x+3}{2026}+\dfrac{x+4}{2027}=4\text{ ?}\)

\(\Rightarrow\text{ }\dfrac{x+1}{2024}+\dfrac{x+2}{2025}+\dfrac{x+3}{2026}+\dfrac{x+4}{2027}-4=0\)

\(\Rightarrow\text{ }\dfrac{x+1}{2024}+\dfrac{x+2}{2025}+\dfrac{x+3}{2026}+\dfrac{x+4}{2027}-1-1-1-1=0\)

\(\Rightarrow\left(\dfrac{x+1}{2024}-1\right)+\left(\dfrac{x+2}{2025}-1\right)+\left(\dfrac{x+3}{2026}-1\right)+\left(\dfrac{x+4}{2027}-1\right)=0\)

\(\Rightarrow\left(\dfrac{x+1-2024}{2024}\right)+\left(\dfrac{x+2-2025}{2025}\right)+\left(\dfrac{x+3-2026}{2026}\right)+\left(\dfrac{x+4-2027}{2027}\right)=0\)

\(\Rightarrow\dfrac{x-2023}{2024}+\dfrac{x-2023}{2025}+\dfrac{x-2023}{2026}+\dfrac{x-2023}{2027}=0\)

\(\Rightarrow\left(x-2023\right)\left(\dfrac{1}{2024}+\dfrac{1}{2025}+\dfrac{1}{2026}+\dfrac{1}{2027}\right)=0\)

Mà \(\dfrac{1}{2024}+\dfrac{1}{2025}+\dfrac{1}{2026}+\dfrac{1}{2027}\ne0\)

\(\Rightarrow x-2023=0\)

\(\Rightarrow x=0+2023\)

\(\Rightarrow x=2023\)

Vậy, \(x=2023.\)

\(\dfrac{55-x}{1963}+\dfrac{50-x}{1968}+\dfrac{45-x}{1973}+\dfrac{40-x}{1978}+4=0\)

\(\Rightarrow\text{ }\dfrac{55-x}{1963}+\dfrac{50-x}{1968}+\dfrac{45-x}{1973}+\dfrac{40-x}{1978}+1+1+1+1=0\)

\(\Rightarrow\text{ }\left(\dfrac{55-x}{1963}+1\right)+\left(\dfrac{50-x}{1968}+1\right)+\left(\dfrac{45-x}{1973}+1\right)+\left(\dfrac{40-x}{1978}+1\right)=0\)

\(\Rightarrow\text{ }\dfrac{2018-x}{1963}+\dfrac{2018-x}{1968}+\dfrac{2018-x}{1973}+\dfrac{2018-x}{1978}=0\)

\(\Rightarrow\text{ }\left(2018-x\right)\left(\dfrac{1}{1963}+\dfrac{1}{1968}+\dfrac{1}{1973}+\dfrac{1}{1978}\right)=0\)

Mà \(\dfrac{1}{1963}+\dfrac{1}{1968}+\dfrac{1}{1973}+\dfrac{1}{1978}\ne0\)

\(\Rightarrow\text{ }2018-x=0\)

\(\Rightarrow\text{ }x=2018-0\)

\(\Rightarrow\text{ }x=2018\)

Vậy, \(x=2018.\)