let 'x' and 'y' represent the two whole numbers, x>y (y is the smaller)

5 times the smaller number added to 3 times the greater number makes 59:

5y + 3x = 59

by solving for y we find:

y = (59 - 3x)/5

click here to see the step by step solution:

for x = 23 we find:

y = (59 - 23*3)/5

y = 2

two such numbers are 23 and 2.