This can facilitate the development of the elimination method as students see that the differences between the left-hand and right-hand sides are 2x and 2 respectively and, thereby, intuitively eliminate the second variable. Some might have difficulty appreciating that the differences are in fact 2x and -2 when calculated formally. 

When the simultaneous equations are formed of three consecutive terms of linear sequences, the solutions will always be x = -1 and y = 2. The proof is accessible to students in secondary school.

Lines of inquiry

Once classes are comfortable with solving simultaneous equations, the inquiry tends to become fast-moving and multi-faceted as individuals or groups work on their own questions. Changes to the prompt have included:

• Descending linear sequences (or one ascending and one descending);

• Using sequences of other types, particularly quadratic and Fibonacci. Why are the solutions for the latter always x = 1 and y = 1?;

• Changing both signs to subtraction or to addition and subtraction;

• Representing the equations on a graph (What is the relevance of the point of intersection?); and

• Using x2 or y2 or both.