What is an algebra problem that can be solved by elimination AND substitution?

Math word problem utilizing the Substitution and Elimination methods:"The sum of Louise's age and Tom's age is 34 yrs. Five years ago, the sum of twice Louise's age and three times Tom's age was 61 yrs. How old are they now?"


Louise = L
Tom = T

Elimination Method:
L + T = 34
2(L - 5) + 3(T - 5) = 61
2L - 10 + 3T - 15 = 61
2L + 3T - 25 = 61
2L + 3T = 86

L + T = 34
2L + 3T = 86
---------------
-3(L + T = 34)
2L + 3T = 86
---------------
-3L - 3T = -102
2L + 3T = 86
---------------
-L = -16 ? L = 16
16 + T = 34 ? T = 18


Substitution Method:
L + T = 34 ? L = 34 - T
2(L - 5) + 3(T - 5) = 61
2L - 10 + 3T - 15 = 61
2L + 3T - 25 = 61
2L + 3T = 86

2(34 - T) + 3T = 86
68 - 2T + 3T = 86
68 + T = 86
T = 18

Currently, Louise is 16 yrs old and Tom is 18 yrs old.

Verify the answers:
L + T = 16 + 18 = 34
2(16 - 5) + 3(18 - 5) = 2(11) + 3(13) = 22 + 39 = 61

Offcourse simultaneous equations can be solved by elimination and substitution.These are algebric equations having more than one unknown variables.(x,y,z,k,... and so on).Here I will show you a simple example of solving a simultaneous equation by both substitution and elimination.

Problem :  Sum of the ages of Tom and Harry is 10,while the difference of their ages is 2.Find their individual ages?

Solution:
                     Let Tom's age be x yrs. and Harry' be y yrs.
                      By given conditions,
                      x+y=10 -----------(1st equation)
                      x-y=2 --------------( 2nd equation)
                     ( By elimination)
                      Add up the 2 equations :
                      L.H.S=(x+y)+(x-y)=2x
                      R.H.S=10+2=12
                      Now,after elimination the equation becomes :
                      2x=12
                      (dividing both sides by 2)
                        x=6;
                        putting x=6 in 1st equation would yield us:
                        y=4(  6-y=2 or,y=6-2or,y=4)
                        (By substitution)
                        From 1st equation,
                        x=10-y;
                        putting the value of x in 2nd equation,we get;
                        (10-y)-y=2
                     or,10-2y=2
                     or,-2y=2-10
                     or,-2y=-8
                      (dividing both sides by -2)
                     or, y=4
                     Again by putting the value of y in 1st equation ,we get;
                     x=6

                     So,by both the methods we can conclude that,
                     age of Tom=6 yrs.
                     age of Harry=4 yrs.

                    NOTE: In simultaneous equations,number of unknown
                               variables must be at least equal or less than  
                               the number of equations.Otherwise,solution is  
                               not possible.