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### Number and algebra

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### Working mathematically

### For younger learners

### Advanced mathematics

# Symmetricality

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### Real(ly) Numbers

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Nurturing young mathematicians: teacher webinars

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Age 14 to 18

Challenge Level

Here is a set of five equations:

$$b+c+d+e=4\\

a+c+d+e=5\\

a+b+d+e=1\\

a+b+c+e=2\\

a+b+c+d=0$$

What do you notice when you add the five equations?

Can you now find the values of $a, b, c, d$ and $e$?

Here is a different set of equations:

$$xy = 1\\

yz = 4\\

zx = 9$$

What do you notice when you multiply the three equations given above?

Can you now find the values of $x, y$ and $z$?

Is there more than one possible set of values?

Here is a third set of equations:

$$ab = 1\\

bc = 2\\

cd = 3\\

de = 4\\

ea = 6$$

Can you find all the sets of values ${a, b, c, d, e}$ that satisfy these equations?

**Extension**

You may like to have a go at Overturning Fracsum.

Can you create your own set of symmetrical equations?

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?