Math Is Fun Forum

What is the ordered pair of real numbers (x,y) for which
123x+ 321y= 345
and

321x+ 123y= 543?

For the areas marked in the figure, T1:T2:T3=2:7:3. What are the ratios of the lengths x to y, and u to v?

In my work I have faced an optimization problem to resolve which I have to solve the following expression

I know that to get  I can use the Omega function Lambert W Function, however, I can't understand how it works. Thus, I ask for help of those of you who know how to solve this kind of equations (showing as much steps as you can) in order to give me reference point for further studying of this topic.

In triangle ABC where AB=4,BC=5,AC=6 Prove that cos 2C=cos B

An insect sets off upwards from the shaft of the minute hand of a church-clock exactly at 12 o'clock. Moving uniformly along the hand, it reaches the end of the hand in a quarter of an hour. When was it at the highest position?

If Tony can cut the trunk of the tree into three pieces in 20 minutes how much time does Tony need to cut the same trunk into six pieces

Given that  is a complex number such that

find

Find a solution to the system of simultaneous equations

Let

with