# Understanding calculating the intercept C between two points

I currently started with some basic geometry and I'm already stuck at some very very basic intuition regarding finding the line between two points in a plane.

I understand $$y = mx + c$$ and I am able to calculate all variables. The way I calculated $$c$$ thus far has been by finding the slope, and use one point in the plane to find the remainder as $$c$$ through $$y = mx + c$$.

Now the textbook used the following points: $$A: (-1, -1)$$ and $$B: (1, 2)$$ which results in $$y = \frac{3}{2}x+\frac{1}{2}$$ and I was able to do this myself by hand.

However, a different method without using one point and a calculcated slope involves using the following equation:

$$c = \frac{x_2y_1 - x_1y_2}{x_2-x_1}$$

But I cannot wrap my head around or find the intuition as to why I am multiplying $$x_2$$ with $$y_1$$ and subtracting $$x_1$$ multiplied by $$y_2$$.

Considering we're dividing by $$x_2 - x_1$$ it must have something to do with the differences in $$y$$. I've calculcated both products but I don't see some sort of relation.

As a test case I used a formula I just came up with: $$y = 3x + 4$$ and took points $$C: (-2, -2)$$ and $$D: (4,16)$$ just to have another example but I am still stuck with why I am doing this and what the products: $$x_2*y_1=4*-2=-8$$ and $$x_1*y_2=-2*16=-32$$ tell me.

$$\frac{24}{6}$$ obviously is $$4$$ which would be the correct $$c$$. Yet I am missing intuition and I really want to understand this. Can someone help me?

• Fact 1: $$y=mx+c$$. We'll rewrite this as $$c=y-mx$$
• Fact 2: $$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ for any points $$(x_{1},y_{1})$$ and $$(x_{2},y_{2})$$ on the line
Now, any point on the line satisfies the equation in fact $$(1)$$; in particular, $$(x_{1},y_{1})$$ satisfies it. Therefore \begin{align} c &= y_{1}-mx_{1} \\ &=y_{1}-\frac{y_{2}-y_{1}}{x_{2}-x_{1}}x_{1} \\ &= \frac{y_{1}(x_{2}-x_{1})-(y_{2}-y_{1})x_{1}}{x_{2}-x_{1}} \\ &=\frac{y_{1}x_{2}-y_{2}x_{1}}{x_{2}-x_{1}} \end{align}
We can rearrange the equation as follows: \begin{align*} y &= mx+c \\ c &= y - mx \\ c &= y - \left(\frac{y_2 - y_1}{x_2 - x1}\right)x \\ c &= \frac{y(x_2 - x_1)}{x_2 - x_1} - \left(\frac{y_2 - y_1}{x_2 - x1}\right)x \\ c &= \frac{yx_2 - yx_1 - xy_2 + xy_1}{x_2 - x_1} \end{align*} Now, since this equation holds true for any pair $$(x,y)$$ on the line, in particular it holds for the point $$(x_2, y_2)$$, so we get \begin{align*} c &= \frac{yx_2 - yx_1 - xy_2 + xy_1}{x_2 - x_1} \\ c &= \frac{y_2 x_2 - y_2 x_1 - x_2 y_2 + x_2 y_1}{x_2 - x_1} \\ c &= \frac{x_2 y_1 - y_2 x_1}{x_2 - x_1} \end{align*}