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Algebra Level 3

If y ( x ) − 1 = 2 x + 5 3 x + 4 \displaystyle{y(x)^{-1}=\frac{2x+5}{3x+4}} , what is y ( x ) y(x) ?

y = − 5 x + 4 2 x − 3 y=-\frac{5x+4}{2x-3} y = 4 x − 5 3 x + 2 y=\frac{4x-5}{3x+2} y = 5 − 4 x 3 x − 2 y=\frac{5-4x}{3x-2} y = − 4 x + 5 3 x − 2 y=-\frac{4x+5}{3x-2}

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1 solution

Will McGlaughlin
Apr 23, 2018
  • let y ( x ) − 1 y(x)^{-1} = y
  • then swap x and y.
  • we get x = ( 2 y + 5 3 y + 4 x=(\frac{2y+5}{3y+4} )
  • x ( 2 y + 5 ) = 3 y + 4 x(2y+5)=3y+4
  • 2 x y + 5 x = 3 y + 4 2xy+5x=3y+4
  • 2 x y − 3 y = 4 − 5 x 2xy-3y=4-5x
  • y ( 2 x − 3 ) = 4 − 5 x y(2x-3)=4-5x
  • y = ( 4 − 5 x 2 x − 3 y=(\frac{4-5x}{2x-3} )
  • y = ( 5 x − 4 3 − 2 x y=(\frac{5x-4}{3-2x} )
  • y ( x ) = 5 x − 4 3 − 2 x y(x)=\boxed{\frac{5x-4}{3-2x}}
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