To be honest, this sounds less like an algebra question... The line of best fit probably means fitting to least squares. That means minimising the total squared distance between the y-co-ordinates of your fitted line, and the true y-values, for each x. It's a very long process to do by hand, but is very easy in a program like MS Excel. For example you can enter the values, create a scatter plot with age on the x-axis and weight on the y-axis, then select the graph and "Add Trendline". Choose linear (you can also tell it to show you the equation, y=mx+c, and the R2, which is a measure of how well it fits from 0 (totally random) to 1 (perfect).
Note that minimising distance is close to the way humans would judge the "fit" by eye - so you can usually get a pretty close line to the actual "best fit" just by trying to draw a straight line through the data yourself!