Sistemi lineari

{3·x·(x + 1) - 2·(y + 1) = 1 + 3·x^2

{1/4·x - 3·(y - 2)/2 = 5·(1 - x) + y + 13/4

Portiamo alla forma normale il sistema

{(3·x^2 + 3·x) - (2·y + 2) = 1 + 3·x^2

{x - 6·y = - 20·x + 4·y + 21

------------------------------

{3·x - 2·y = 3

{21·x - 10·y = 21

Lo risolviamo con Cramer e per Sostituzione (gli altri due metodi sono già stati fatti da @eidosm)

[x = Δx/Δ = 1 ∧ y = Δy/Δ = 0]

----------------------------

y = 3·(x - 1)/2  dalla prima

21·x - 10·(3·(x - 1)/2) = 21----> x = 1

y = 3·(1 - 1)/2----> y = 0

Interpretazione grafica:

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$\small \begin{Bmatrix}
3x(x+1)-2(y+1)&=&1+3x^2\\
\dfrac{1}{4}x-3·\dfrac{y-2}{2}&=&5(1-x)+y+\dfrac{13}{4}\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
3x^2+3x-2y-2&=&1+3x^2\\
\dfrac{1}{4}x-3·\dfrac{y-2}{2}&=&5-5x+y+\dfrac{13}{4}\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
\cancel{3x^2}+3x-2y\cancel{-3x^2}&=&1+2\\
x-6(y-2)&=&20-20x+4y+13\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
3x-2y&=&3\\
x-6y+12&=&33-20x+4y\\
\end{Bmatrix}$ $^{(1)}$

$\small \begin{Bmatrix}
\dfrac{\cancel3x}{\cancel3}-\dfrac{2}{3}y&=&\dfrac{\cancel3^1}{\cancel3_1}\\
x-6y+20x-4y&=&33-12\\
\end{Bmatrix}$ 

$\small\text{procedi con sostituzione:}$

$\small \begin{Bmatrix}
x-\dfrac{2}{3}y&=&1\\
21x-10y&=&21\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1+\dfrac{2}{3}y\\
21\left(1+\dfrac{2}{3}y\right)-10y&=&21\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1+\dfrac{2}{3}y\\
21+14y-10y&=&21\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1+\dfrac{2}{3}y\\
4y&=&21-21\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1+\dfrac{2}{3}y\\
4y&=&0\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1+\dfrac{2}{3}y\\
\dfrac{\cancel4y}{\cancel4}&=&\dfrac{0}{4}\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1+\dfrac{2}{3}y\\
y&=&0\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1+\dfrac{2}{3}·0\\
y&=&0\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1+0\\
y&=&0\\
\end{Bmatrix}$

$\small \begin{Bmatrix}
x&=&1\\
y&=&0\\
\end{Bmatrix}$

$\small\text{dal punto}$ $ ^{(1)}\small\text{procedi anche con il metodo Cramer:}$

$\small  \begin{Bmatrix}3x-2y&=&3\\x+20x-6y-4y&=&33-12\\\end{Bmatrix}$

$\small \begin{Bmatrix}3x-2y&=&3\\21x-10y&=&21\\\end{Bmatrix}$

$\small \text{determinante D } =\begin{vmatrix}
3&-2\\
21&-10\\
\end{vmatrix}$ $\small = (21·-2)-(3·-10) \Longrightarrow -42-(-30) \Longrightarrow -42+30 = -12$

$\small \text{determinante Dx } =\begin{vmatrix}
3&-2\\
21&-10\\
\end{vmatrix}$ $\small = (21·-2)-(3·-10) \Longrightarrow -42-(-30) \Longrightarrow -42+30 = -12$

$\small \text{determinante Dy } =\begin{vmatrix}
3&+3\\
21&+21\\
\end{vmatrix}$ $\small = (21·3)-(3·21) \Longrightarrow 63-63 = 0$

$\small x= \dfrac{D_x}{D} = \dfrac{-12}{-12} = 1$

$\small y= \dfrac{D_y}{D} = \dfrac{0}{-12} = 0$

 $\small\text{grafico da WolframAlpha.}$

Grafico 

https://www.desmos.com/calculator/78edtijonz