### 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

### Bang's Theorem

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

### Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

# Negatively Triangular

##### Stage: 4 Challenge Level:

Patrick from Woodbridge school used a computer to check his answer:

"I used the Mac OS X application "Grapher" to help with this. I used it to plot the lines to check my answers. I did the intersections part by solving every equation against every other in a simultaneous equation, I got the answer of $6$ (I did each algebraically but it would take a lot of space to write up)"

Labelling the equations $1$ to $4$ in the obvious way, here are Patrick's co-ordinates of the six intersections:

$1$ and $2$: $(9,4)$
$1$ and $3$: $(1,7)$
$1$ and $4$: $(-47,25)$
$2$ and $3$: $(-1,-1)$
$2$ and $4$: $(-5,-3)$
$3$ and $4$: $(-2,-6)$

We can deduce that lines $2, 3$ and $4$ form the triangle.

Note that if we rearrange the equations intothe standard form $y = mx + c$, it would be possible to sketch the four lines and estimate roughly where their intersections lie. It might then be sufficient to solve the problem without needing to explicitly calculate any co-ordinates.