Solve the following system of equation: #[((1), sqrt(2)x+sqrt(3)y=0),((2), x+y=sqrt(3)-sqrt(2))]#?

1 Answer
Apr 19, 2016

Answer:

#{(x = (3sqrt(2)-2sqrt(3))/(sqrt(6)-2)), (y = (sqrt(6)-2)/(sqrt(2)-sqrt(3))):}#

Explanation:

From #(1)# we have

#sqrt(2)x+sqrt(3)y = 0#

Dividing both sides by #sqrt(2)# gives us

#x + sqrt(3)/sqrt(2)y = 0" (*)"#

If we subtract #"(*)"# from #(2)# we obtain

#x+y-(x+sqrt(3)/sqrt(2)y) = sqrt(3)-sqrt(2) - 0#

#=> (1-sqrt(3)/sqrt(2))y = sqrt(3)-sqrt(2)#

#=> y = (sqrt(3)-sqrt(2))/(1-sqrt(3)/sqrt(2))=(sqrt(6)-2)/(sqrt(2)-sqrt(3))#

If we substitute the value we found for #y# back into #"(*)"# we get

#x + sqrt(3)/sqrt(2)*(sqrt(6)-2)/(sqrt(2)-sqrt(3)) = 0#

#=> x + (3sqrt(2)-2sqrt(3))/(2-sqrt(6)) = 0#

#=> x = -(3sqrt(2)-2sqrt(3))/(2-sqrt(6)) = (3sqrt(2)-2sqrt(3))/(sqrt(6)-2)#

Thus, we arrive at the solution

#{(x = (3sqrt(2)-2sqrt(3))/(sqrt(6)-2)), (y = (sqrt(6)-2)/(sqrt(2)-sqrt(3))):}#