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?

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.

Intersections

Stage: 4 and 5 Challenge Level:

Why do this problem?

On the one hand this problem is a simple exercise in solving pairs of linear simultaneous equations but on the other it provides perhaps unexpected results that call for investigating the connection between the algebra and geometry, and considering the equations of the lines and gradients.

Possible approach
Set this as homework or as a lesson starter and have a class discussion about the results.

Key questions
Why are the solutions of the two pairs of simultaneous equations so different when the equations are so nearly the same?