### 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

### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

### 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

### Why do this problem?

This problem provides a good opportunity to explore simultaneous linear equations, both graphically and algebraically.

### Possible approach

This printable worksheet may be useful: Negatively Triangular.

Start by discussing the question "How many intersections do you expect there to be?" This could be broken down by considering how many intersections two lines would have, then three, then four.

Emphasise that we won't know for sure until we consider the graphs or solve the simultaneous equations, as there may be a pair of lines which don't intersect at all, or three lines which intersect at a single point.

Then set the main challenge - which three lines enclose a triangle which lies in the third quadrant; that is, all of the points in the triangle have a negative x and a negative y coordinate.

Students might choose to sketch the four graphs to get some insight into where the lines intersect. Alternatively they might solve each pair of simultaneous equations to find all six points of intersection. Some students might find it easier to work with the equations in the form $y=mx+c$.

After students have worked on the problem, you may wish to show them the Desmos graph linked from the solution page; this is a good opportunity to show the power of graphing software for gaining insight into graphical situations.

### Key questions

What is the gradient of each line? Are any of the lines parallel?
Where does each line cross the axes?
Does it help to draw a rough sketch of each line?
How can you use algebra to find out the exact coordinates of the points of intersection?

### Possible support

Which is Cheaper? provides a real-life context for exploring equations of straight lines and graphical interpretations of simultaneous equations.

### Possible extension

Students might like to consider making up similar problems of their own, perhaps choosing lines which intersect in one of the other quadrants.

They could also explore Which is Bigger? which also invites students to create sets of equations whose intersections satisfy certain constraints.