Shortcut trick to solve linear equations in 2 variables

It sucks, right When you're in the middle of some long question, and you have to solve a system of linear equations in 2 variables to move forward? This post is dedicated for a simple formula/shortcut trick to solve it within a few seconds. Let's get right to it.

$\\ \text{Given equations,}\\ \text{A}_1X +\text{B}_1Y = C_1\\ \text{A}_2X +\text{B}_2Y = C_2\\$

To solve for X, you just have to use this simple formula,

$X = \frac{-(B_1.C_2-B_2.C_1)}{A_1.B_2-B_2.A_1}$

and to find the value of y, you can simply put the value of x in any equation, or you could simply use the formula,

$Y = \frac{A_1.C_2-A_2.C_1}{A_1.B_2-B_2.A_1}$

Yeah, I know, the formulas seem very complicated, but trust me, it becomes very easy once you get the hang of it. To remember it, just know that we're just multiplying the diagonals, and subtracting them, you could also call this, "CRISS CROSS METHOD".
Maybe, the derivation of this shortcut will help you understand it better, so let's get it to it.

$\\ \text{Given equations,}\\ \text{A}_1X +\text{B}_1Y = C_1\\ \text{A}_2X +\text{B}_2Y = C_2\\$

$\\ \text{We can represent the following with the help of matrices}\\ \begin{bmatrix} A_1 &B_1 \\ A_2 &B_2 \end{bmatrix}\times \begin{bmatrix} X\\ Y \end{bmatrix}=\begin{bmatrix} C_1\\ C_2 \end{bmatrix} \\\\ \begin{bmatrix} X\\ Y \end{bmatrix}=\begin{bmatrix} A_1 &B_1 \\ A_2 &B_2 \end{bmatrix}^{-1}\times\begin{bmatrix} C_1\\ C_2 \end{bmatrix}\\\\ \begin{bmatrix} X\\ Y \end{bmatrix}= \frac{1}{det(A)} \begin{bmatrix} B_2 &-B_1 \\ -A_2 &A_1 \end{bmatrix}\times\begin{bmatrix} C_1\\ C_2 \end{bmatrix} \\\\ \begin{bmatrix} X\\Y \end{bmatrix} = \frac{1}{A_1.B_2-B_1.A_2}\begin{bmatrix} B_2.C_1& -B_1.C_2\\ -A_2.C_1& A_1.C_2 \end{bmatrix} \\ \\ \therefore \text{ } X = \frac{B_2.C_1-B_1.C_2}{A_1.B_2-B_1.A_2},\text{and, } Y = \frac{A_1.C_2-A_2.C_1}{A_1.B_2-B_1.A_2}$

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Shortcut trick to solve linear equations in 2 variables

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