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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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Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

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Pair Squares

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

System Speak

Stage: 5 Challenge Level: Challenge Level:1

Why do this problem?

When working on simultaneous equations, it's good to see non-standard examples like this one. Although it is no more difficult than a standard simultaneous equations problem, the unfamiliarity requires students to think creatively.

Possible approach

Display the system of equations:

$ab = 1$
$bc = 2$
$cd = 3$
$de = 4$
$ea = 6$
"Here are five equations with 5 unknowns.
Can you find values for a, b, c, d and e that solve all five equations?"

Give students some time to work with a partner to try possible approaches. They may use trial and improvement, or they may use substitution and elimination.

Once students have had time to tackle the problem, share approaches.

One rather cunning method is to multiply all five equations together, to give $(abcde)^2 = 144$, so $abcde = \pm12$, and then divide by pairs of equations to find each letter. If no-one comes up with this method, you may wish to show it to them.

Finally, it is worth discussing that there are two solution sets, as the values may all be negative!

Key questions

Can you find an expression for b in terms of a?
Can you find an expression for c in terms of b?
Can you combine these to give an expression for c in terms of a?

Possible extension

Students may find Intersections a thought-provoking challenge on simultaneous equations.

Possible support

Multiplication Arithmagons invites students to solve a similar system in a context, with fewer variables.