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HELM Book 1.5.1: using formulae and substitution: task 1

Add two algebraic fractions with coprime denominators. Part of HELM Book 1.4.3.

HELM Book 1.4.3 Exercises: addition and subtraction of algebraic fractions

There are two parts: 

(3x)/4-x/5+x/3 and (3x/4)-(x/5+x/3).

The numbers are randomised to small, coprime, positive integers.

This is a fixed, unmarked, question:

Show that (x1)/((1/x3)-(1/x2))) = (x1x2x3)/(x2-x3)

Given 2 or 3 fractions such as A/(2x+5), add them.

There are 5 possible versions.

Add (a/b).x +/- (c/d) where a,b,c,d are randomised positive integers, and x is a randomised letter.

Simplify (qx+a)/(rx+b) +/- (sx+c)/(rx+b)^2

x is a randomised variable. a,b,c,d,q,r are randomised integers.

Simplify (qx+a)/(rx+b) +/- (sx+c)/(tx+d)

x is a randomised variable. a,b,c,d,q,r,s,t are randomised integers. a,b,c,d run from -5 to 5, including 0. q,r,s,t run from -3 to 3, and can be 0 if the constant term is nonzero, but are mostly 1.

Simplify ax/b +/- cx/d, where x is a randomised variable, and a,b,c,d are randomised integers.