The sum of two numbers is 83. if twice the larger number is added to three times the smaller number, the result is 197. find the numbers.

THE LARGER NUMBER IS 52, WHILE THE SMALLER NUMBER IS 31.

solution 
let assume the smaller to be y 
let assume the larger number to be x

x + y = 83 1
2x + 3y = 1972

using simultaneous equation
make x the subject of formular in equation 1

x=83 - y 3

substitute equation 3 in equation 2
any where you see x replace it with 83-y in equation 2
2(83 - y) + 3y = 197
open the bracket by multiplying by 2 by the content of the bracket
166 - 2y + 3y = 197
-2y +3y = y
therefore:
166 + y = 197
collect like terms by taking 166 to the lefthand side of the equation
y = 197 - 166
y = 31.

substitute y in equation 2 above
x + 31 = 83
collect like terms by taking 31 to the lefthand side of the equation
x = 83 - 31
x = 52

X represent the larger number which is 52
Y represent the lsmaller number which is 31