A discrete, linear graph at 45 degrees, where the N points add up to 1

Question

I have a question:

I have a set of points that represent a graph (x0,x1..x9) Lets say 10 points. They are at a linear 45 degree angle up (Gradient 1). I am also told that (x0+x1+x2..x9 = 1).

How can I solve for each xi? Can I get an idea of how to start?

Answer 1

Telling us that you have a linear graph seems to imply that $x$ is actually a function of an independent variable, and one should read $x(i)$ for $x_i$.

Then $x(i)$ must be a linear function with slope $1$, so that $x(i)=i+C$.

Using the given constraint, $$(0+C)+(1+C)+\dots(9+C)=45+10C=1,$$ and $$C=-\frac{44}{10}.$$

Alternatively, if your data points are couples $(x_i,y_i)$ forming a straight line with unit slope, then all you can say is $$y=x+C,$$ and $$\sum_{i=0}^9 y_i=1+10C,$$ but you can't guess $C$.