How does $\sum x_i y_i = \bar{x}\sum y_i$?

Question

This is the identity given in my textbook with regards to the centre of mass of of a set of particles along the line. $x$ could be the distance along the line, and $y$ could be the mass. $\sum x_i y_i$ would then be the sum of moments of forces.

Answer 1

This is the definition of the centre of mass; more specifically, the centre of mass $\bar x$ is defined in this context by $$ \bar x = \bigg( \sum x_i y_i \bigg) \bigg/ \sum y_i $$ (the horizontal moment divided by the mass).