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If you put the line equations in the form  y = mx + b   the slope is m

First one  y -ax =2

                y = ax + 2        the slope of this line is 'a'

                                       perpindicular slope will be  - 1/a

Line 2  x + 23y = 138

                 23y = -x + 138    divide both sides by 23

                     y = -1/23  x  + 6       slope is -1/23     

                         for these lines to be perpindicular:    - 1/a  = -1/23        so a = 23

Here is a graph:

https://www.desmos.com/calculator/fqvxr43leu

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