Question 779636
{{{x=y/sqrt(1+y^2)}}}
Expressing y in terms of x means to have an equation in the form:
y = expression with x in it
(or expression with x in it = y)
This will require that y not be in a denominator or in a square root. So we will eliminate the fraction first then the square root and then we will see what needs to happen next.<br>
Multiplying both sides by the square root (i.e. the denominator) will eliminate the fraction:
{{{x*sqrt(1+y^2)=y}}}
With just one term on the left side the square root is isolated (which is required to eliminate it). So we can eliminate the square root by squaring both sides:
{{{(x*sqrt(1+y^2))^2=(y)^2}}}
{{{x^2*(1+y^2)=y^2}}}
{{{x^2+x^2*y^2=y^2}}}<br>
Now we solve for y. First we gather the y terms on one side of the equation. Subtracting {{{x^2*y^2}}} from each side:
{{{x^2=y^2-x^2*y^2}}}
Since the terms on the right side are not like terms we cannot combine them. But we can factor out the {{{y^2}}}:
{{{x^2=y^2(1-x^2)}}}
Dividing both sides by {{{1-x^2}}}:
{{{x^2/(1-x^2)=y^2}}}
And finally we find the square root of each side (<u>remembering both square roots</u> (positive and negative)):
<u>+</u>{{{sqrt(x^2/(1-x^2))=y}}}