document.write( "Question 779636: What are the steps required to express y in terms of x if x=y/sqrt(1+y^2) \n" ); document.write( "

Algebra.Com's Answer #475306 by jsmallt9(3758) $\"\"$ $\"About$

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$\"x=y%2Fsqrt%281%2By%5E2%29\"$
\n" ); document.write( "Expressing y in terms of x means to have an equation in the form:
\n" ); document.write( "y = expression with x in it
\n" ); document.write( "(or expression with x in it = y)
\n" ); document.write( "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.

\n" ); document.write( "Multiplying both sides by the square root (i.e. the denominator) will eliminate the fraction:
\n" ); document.write( " $\"x%2Asqrt%281%2By%5E2%29=y\"$
\n" ); document.write( "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:
\n" ); document.write( " $\"%28x%2Asqrt%281%2By%5E2%29%29%5E2=%28y%29%5E2\"$
\n" ); document.write( " $\"x%5E2%2A%281%2By%5E2%29=y%5E2\"$
\n" ); document.write( " $\"x%5E2%2Bx%5E2%2Ay%5E2=y%5E2\"$

\n" ); document.write( "Now we solve for y. First we gather the y terms on one side of the equation. Subtracting $\"x%5E2%2Ay%5E2\"$ from each side:
\n" ); document.write( " $\"x%5E2=y%5E2-x%5E2%2Ay%5E2\"$
\n" ); document.write( "Since the terms on the right side are not like terms we cannot combine them. But we can factor out the $\"y%5E2\"$ :
\n" ); document.write( " $\"x%5E2=y%5E2%281-x%5E2%29\"$
\n" ); document.write( "Dividing both sides by $\"1-x%5E2\"$ :
\n" ); document.write( " $\"x%5E2%2F%281-x%5E2%29=y%5E2\"$
\n" ); document.write( "And finally we find the square root of each side (remembering both square roots (positive and negative)):
\n" ); document.write( "+ $\"sqrt%28x%5E2%2F%281-x%5E2%29%29=y\"$ \n" ); document.write( "

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