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Given:#" "ax + by =c#

Subtract #color(red)(ax)# from both sides

#color(green)(ax color(red)(-ax) + by" " =" "c color(red)(-ax)#

#color(green)(0+by" "=" "c-ax)#

Divide both sides by #color(red)(b)#

#color(green)((by)/(color(red)(b))" "=" "c/(color(red)(b))-(ax)/(color(red)(b))#

But #b/b=1#

#color(green)(y" "=" "c/b-(ax)/b)#

Changing the order to match convention we have:

#color(green)(y=-a/b x+c/b)#

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Compare #y=-a/b x + c/b" "# to #" "y=4x-7#

So we have:

#-a/b=4" " ->" " a/b=-4" "->" "a=-4b#

#c/b=-7" "->" "c=-7b#

This gives us a sort of 'open book' about choices for #a and c#

Example: Let #b=3# or any value you so choose, then we have:

#a=-4(3)=-12#

#c=-7(3)=-21#

Thus #ax+by=c " implies that "-12x+3y=-21#

#3y=+12x-21#

#y=12/3x-21/3 -> 4x-7 color(red)(" IT WORKS!")#