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

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Answer 1 · 2016-04-19T00:04:08

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

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

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

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