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Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

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Overturning Fracsum

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

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Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

System Speak

Stage: 4 and 5 Challenge Level: Challenge Level:1

Why use this problem?

The problem gives practice in the techniques for solving simultaneous equations. As a non-standard problem, it is designed to call 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 pairs 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 it gives practice in communication of mathematical ideas.

Key question

(only if the learners are really at a loss as to what to do)
Can you eliminate the variable between pairs of equations?