From the condition, you have the system of 2 equations in 2 unknowns

13*d + 14*p = 233,    (1)    (Anjali' spending)
 5*d +  7*p = 109.    (2)    (Joe' spending)


To solve the system, multiply equation (2) by 2 (both sides). The modified system is

13*d + 14*p = 233,    (3)   
10*d + 14*p = 218.    (4) 


Now subtract eq(4) from eq(3) (both sides).  The terms "14*p" in both equations will cancel each other, and 
you will get a single equation for only one unknown "d" (it is how the Elimination method works):

3d = 233 - 218  = 15  ====>  d =   = 5.


Thus you found that the pot of ivy costs $5.


Then from eq(2),   7p = 109 - 5*5 = 84  and  d =  = 12.


Answer.  One daylily costs $12  and  one pot of ivy costs $5.