Move expression to the left side and change its sign
#5/(y-3)+10/(y^2-y-6)-y/(y+2)=0#
Write #-y# as a sum or difference
#5/(y-3)+10/(y^2+2y-3y-6)-y/(y+2)=0#
Factor out #y# and #-3# from the expression
#5/(y-3)+10/(y(y+2)-3(y+2))-y/(y+2)=0#
Factor out #y+2# from the expression
#5/(y-3)+10/((y+2)(y-3))-y/(y+2)=0#
Write all numerators above the least common denominator
#(5(y+2)+10-y(y-3))/((y+2)(y-3))=0#
Distribute #5# and #-y# through the parenthesis
#(5y+10+10-y^2+3y)/((y+2)(y-3))=0#
Collect the like terms
#(8y+20-y^2)/((y+2)(y-3))=0#
Use the commutative property to reorder the terms
#(-y^2+8y+20)/((y+2)(y-3))=0#
Write #8y# as a sum or difference
#(-y^2+10y-2y+20)/((y+2)(y-3))=0#
Factor out #-y# and #-2# from the expression
#(-y(y-10)-2(y-10))/((y+2)(y-3))=0#
Factor out #-(y-10)# from the expression
#(-(y-10)(y+2))/((y+2)(y-3))=0#
Reduce the fraction with #y+2#
#-(y-10)/(y-3)=0#
Determine the sign of the fraction
#-(y-10)/(y-3)=0#
Simplify
#(10-y)/(y-3)=0#
When the quotient of expressions equals #0#, the numerator has to be #0#
#10-y=0#
Move the constant, #10#, to the right side and change its sign
#-y=-10#
Change the signs on both sides of the equation
#y=10#
Check if the solution is in the defined range
#y=10, y!=3,y!=-2#
#:. y=10#
