My Formula for slope-intercept Form of a Line

Goal: given two points $(x_1,y_1)$ and $(x_2,y_2)$ , find the straight line in $\R^2$ that connects these two points in the slope-intercept form in terms of the $x_1$ , $y_1$ , $x_2$ and $y_2$ .

$y=mx+c$

$m=\frac{\Delta y}{\Delta x}=\frac{y_1-y_2}{x_1-x_2}$

$\begin{aligned} y&=mx+c\\ y_1&=mx_1+c\\ c&=y_1-mx_1\\ &=y_1-\left(\frac{y_1-y_2}{x_1-x_2}\right)x_1\\ &=\frac{x_1y_1-x_2y_1}{x_1-x_2}-\frac{x_1y_1-x_1y_2}{x_1-x_2}\\ &=\frac{x_1y_1-x_2y_1-x_1y_1+x_1y_2}{x_1-x_2}\\ &=\frac{x_1y_2-x_2y_1}{x_1-x_2}\\ c&=\frac{1}{x_1-x_2}\begin{vmatrix}x_1&y_1\\x_2&y_2\end{vmatrix} \end{aligned}$

$\therefore\boxed{y=\left(\frac{y_1-y_2}{x_1-x_2}\right)x+\frac{1}{x_1-x_2}\begin{vmatrix}x_1&y_1\\x_2&y_2\end{vmatrix}}$

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$