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## 'Always Two' printed from http://nrich.maths.org/

### Why use this
problem?

The problem gives practice in writing equations from verbal
information and in algebraic manipulation. Learners will experience
the value of recognising and making use of the symmetries in the
algebraic expressions that occur.

As a non-standard problem, it is designed for learners to think for
themselves but it does not require any mathematical knowledge
beyond knowing how to solve two linear simultaneous equations in
two unknowns.

### Possible approach

Encourage learners to work in small groups to discuss how they
might tackle the problem, then to work out the solutions
individually, and finally to check together if their answers
agree.This is reassuring for people who are inclined to panic at
the unfamiliar and gives practice in communication of mathematical
ideas.

Learners may find one or both of the solutions by trial and error
but then they need to prove that there are no other
solutions.

### Key questions

Are you using the symmetry of the expressions?

If you subtract one equation from another in pairs, what do you
notice?

What can you deduce if the product of two linear factors in an
equation is zero?

### Possible support

In order to get some experience of thinking for themselves, and not
simply following set procedures to solve a system of equations, the
class could first try the problem

System Speak which is another non-standard problem on
simultaneous equations. System Speak is easier in so far as it can
be solved by expressing all the letters in terms of one of the
letters (eliminating the other variables) and reaching a final
equation in one variable.

As practice in solving more standard sets of simultaneous equations
with unit coefficients in more than 2 unknowns learners could try

Equation Sudoko.
### Possible Extension

Try

Leonardo's Problem where you have first to create the equations
then to solve them.