On a balance scale, four spools and one thimble balance twenty nine buttons. also, one spool balances one thimble and one button. how many buttons will balance one spool

Here are the definitions I just made up:

   S = weight of 1 spool
   B = weight of 1 button
   T = weight of 1 thimble

The question tells us that

   4S + 1T  =  29B
and  
   1S  =  1T + 1B  

Take the second equation, and subtract 1B from each side:

   1T  =  1S - 1B

Take this back to the first equation.  In there, where it says  '1T',
substitute  '1S - 1B'  in place of  1T .

     4S + 1T  =  29B

     4S + 1(1S-1B) = 29B

Eliminate the parentheses:    4S + 1S - 1B = 29B

Add  1B  to each side:            4S + 1S  =  29B + 1B

Combine like terms on the left side:   5S = 29B + 1B

Combine like terms on the right side:   5S = 30B

Divide each side by  5 :                         S  =  6B

Six (6) buttons will balance one (1) spool on the balance scale.

The household makes $424.45