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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.

Always Two

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.