I see you have 2 variables reversed.  You have Tanya's age as y,
and Brian's age as z, but in the equations you have switched these.
Your equations are correct if you let Tanya's age be z and Brian's
age as y.  

To keep from confusing letters, when the people's initials are all 
different, use their initial to represent their age:

The sum of Aaron, Bryan and Tanya's ages is 90 years.

A + B + T = 90

Aaron is 7 years older than Tanya 

A = T + 7

and Bryan is 2 years older than Tanya. 

B = T + 2

So the system is:

A + B + T = 90
A = T + 7
B = T + 2

Write the three equations so that the letters line up vertically:

(1)       A + B + T = 90
(2)       A     - T =  7
(3)           B - T =  2

Multiply eq. (2) by -1 and add eq. (1) to it

         -A     +  T = -7
(1)       A + B +  T = 90
          ---------------
(4)           B + 2T = 83

Multiply eq. (3) by -1 and add eq. (4) to it

             -B +  T = -2
(4)           B + 2T = 83
             ------------
                  3T = 81
                   T = 27

Substitute 27 for T is eq. (3)

              B - 27 =  2
                   B = 29 
                   

Substitute 27 for T and 29 for B in eq. (1)

           A + B + T = 90
         A + 29 + 27 = 90
              A + 56 = 90
                   A = 34

Aaron is 34, Bryan is 29 and Tanya is 27           

Edwin