Frame the equation for 'x' raised to the power 'y' added to 'y' raised to the
power 'x' is equal to one hundred. Solve for 'x' and 'y'. [Hint: 'x', and 'y' are integers and
use laws of indices.]
Sol:
From the statement given, xy + yx = 100 ...(i)
This is an equation in two unknowns 'x' and 'y'. Logically, two equations are required to solve two unknowns. But we just have one.
Let's assume x = 1. Substituting 'x' in (i), we have 1y + y1 = 100 ... (ii)
From the laws of indices, we know:
a) One raised to the power of any integer is one. i.e., 1a = 1.
b) Any number raised to the power of one is the number itself. i.e., a1 = a.
Therefore (ii) becomes, 1 + y = 100 or y = 99.
∴ x = 1, y = 99 is the solution!
(If we assume y = 1, then x = 99 is also the solution).
In general, any natural number "N" can be expressed as
N = xy + yx
where x = 1 and y = N – 1
Note: There is another set of solutions to xy + yx = 100. Can you identify it?
Another set of solutions is: x = 2 and y = 6 (or x = 6 and y = 2).