Question 698137
If he sold 11 boxes,
{{{ 5a + 12b = 99 }}}
{{{ a + b = 11 }}}
and
{{{ 5a + 5b = 55 }}}
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Do the subtraction
{{{ 5a + 12b = 99 }}}
{{{ -5a - 5b = -55 }}}
{{{ 7b = 44 }}}
{{{ b }}} is a fraction, so doesn't work
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{{{ 5a + 12b = 99 }}}
{{{ a + b = 13 }}}
{{{ 5a + 5b = 65 }}}
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{{{ 5a + 12b = 99 }}}
{{{ -5a - 5b = -65 }}}
{{{ 7b = 34 }}}
This is still a fraction
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I notice that in each case,
99 - 5*( number of boxes ) must be divisible by 7
99 - 5*14
99 - 70 = 29 -no
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99 - 5*15 
99 - 75 = 24 -no
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99 - 5*16
99 -80 = 19 -no
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99 - 5*17
99 - 85 = 14 -yes
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{{{ 5a + 12b = 99 }}}
{{{ a + b = 17 }}} 
{{{ 5a + 5b = 85 }}}
Do the subtraction
{{{ 5a + 12b = 99 }}}
{{{ -5a - 5b = -85 }}}
{{{ 7b = 14 }}}
{{{ b = 2 }}}
and, since
{{{ a + b = 17 }}} 
{{{ a = 15 }}}
The storekeeper used 15 of the boxes of 5
and 2 of the boxes of 12
check:
{{{ 5a + 12b = 99 }}}
{{{ 5*15 + 12*2 = 99 }}}
{{{ 75 + 24 = 99 }}}
{{{ 99 = 99 }}}
OK