Tricks for manipulating this equation type.
#color(red)("Very important "->" What you do to one side of the equation you do to the other side.")#
#color(green)("To move an add or subtract to the other side of the = turn it into 0")#
#color(green)("To move a multiply or divide to the other side of the = turn it into 1")#
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Given:#" "(3/x)+(2/m)=(1/n)#
Write as:
#" "3/x+2/m=1/n#
We need to determine the value of #x# so we need to 'get rid' of #2/m# so that we have #3/x# on its own.
Subtract #color(blue)(2/m) " "underline("from both sides")#
#" "color(brown)(3/x+2/m color(blue)(-2/m)" "=" "1/n color(blue)(-2/m))#
#color(white)(.)#
#" "3/x+0=1/n-2/m#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
To do the next bit we need to combine #1/n-2/m#
Multiply #1/n# by 1 but in the form of #1=m/m#
Multiply #2/m# by 1 but in the form of #1=n/n#
#" "3/x=(1/nxxm/m)-(2/mxxn/n)#
#" "3/x=((1xxm)/(nxxm))-((2xxn)/(mxxn))#
#" "3/x=((m)/(mn))-((2n)/(mn))#
#" "3/x=(m-2n)/(mn)#
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Now you can turn everything upside down
#" "x/3=(mn)/(m-2n)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Multiply both sides by #color(blue)(" "3)#
#color(brown)(" "x xx(color(blue)(3))/3=(color(blue)(3)mn)/(m-2n))#
But #3/3 = 1# giving:
#" "x=(3mn)/(m-2n)#
