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Solving simultaneous equations inquiry

Students might require a structured approach to the prompt. They can quickly work out that x = -1, but often overlook the key feature of the prompt - that is, the three numbers in each equation are consecutive terms of a linear sequence. In the case of the prompt, we see 5, 7 and 9 and 3, 7 and 11. The teacher might start with the prompt in the form below to emphasise the connection between the numbers.
In the prompt, the coefficient of y is the same in both equations to facilitate the use of the elimination method. 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 have difficulty appreciating that the differences are in fact 2x and -2 when calculated formally. If the teacher wishes to make the prompt more challenging (perhaps because the class is already aware of the elimination method), then the prompt might have different coefficients of y.
 
When the simultaneous equations are formed of three consecutive terms of linear sequences, the solutions will always be x = -1 and y = 2. (A proof of this is given in the mathematical notes.) Students can verify this common solution for simultaneous equations of the type in the original prompt (in which the coefficients of y 
are the same) through forming and solving their own examples. When they make up examples in which the coefficients of y are different, many classes have requested a teacher's explanation of how to find a solution. 
    
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.

Resources
PowerPoint

Classroom inquiry 
Craig Barton, TES adviser for secondary maths, tried out the inquiry with his year 10 class. He tweeted about the lesson (below) and posted a picture of the approach taken by one student. She inquired into sequences that use the recurrence relation for the Fibonacci sequence: xn = xn-1 + xn-2. Her theory states that simultaneous equations using any three consecutive numbers in such a sequence give the solution x = 1 and y = 1.

You can follow Craig on twitter at @tesMaths. He also runs his own website.