Properties of Equality Relation Let a , b , c  Î  Z ( 1 ) If a = b then a + c = b + c          and if a + c = b + c , then a = b          i. e. We can add or cancel any integer from the two sides of the equality. ( 2 ) If a =b then a c = b c          and if a c = b c , then a = b        i. e. We can multiply both sides of an equality by any integer , or we can cancel any non - zero integer , when it is multiplied in both sides of an equality . Example ( 2 )  Find the solution set of the following equations in   Z : ( 1 ) X +  4 = 3     ( 2 )  X - 5 = 0    ( 3 ) 2X  + 7 = 5 Solution ( 1 ) .:  X + 4 = 3 [ by adding ( -4 ) to both sides ]        \  X  + 4 + (-4 ) = 3 + (-4 )        \   X + 0 = -1    \ X  = -1        \ The solution set is { -1 }  \ S. S. = { -1 } ( 2 ) .: X - 5 = 0 [ by adding ( 5 ) to both sides ]         \  X  - 5 + (5 ) = 0 + (5 )          \   X + 0 = 5   \ X  = 5          \ The solution set is { 5 }  \ S. S. = { 5 } ( 3 ) .: 2X + 7 = 5 [ by adding ( -7 ) to both sides ]         \  2X  + 7 + (-7 ) = 5 + (-7 )          \  2 . X  = -1 . ( 2 ) [ Cancel 2 from both sides ]         \ X  = -1 \ S. S. = { -1 } Exercise : Find the solution set of equation : 3X + 2 = 11 Solution       .: 3X + 2 = 11    [By adding to both sides]     \ 3X + 2 + = 11 +     \ 3X = 9                 \  3 . X  = 3 . ( 3 ) [ Cancel from both sides ]       \ X  = 3 \ S. S. = { } examples