Two New Variables
Date: 10/25/2002 at 13:33:10 From: Robert Summers Subject: Two equations with two unknowns I am trying to calculate using the Owens-Wendt equation. The equation will give two equations with two unknowns. I am unable to solve them, because of the x^1/2 and y^1/2 powers. Two equations: 69.7 = (21.8x)^1/2 + (51.0y)^1/2 43.9 = (49.5x)^1/2 + (1.3y)^1/2 I was thinking of trying to solve by "squaring" each side, but that will still leave x or y with 1/2 exponents. Thanks.
Date: 10/25/2002 at 13:44:07 From: Doctor Achilles Subject: Re: Two equations with two unknowns Hi Robert, Thanks for writing to Dr. Math. I can think of two ways to approach this problem. The first is to square both sides twice. So for example, on the first equation, you will end up with: 69.7^2 = 21.8x + 51.0y + [(21.8*51.0)xy]^1/2 Then you can subtract 21.8x and 51.0y from both sides: 69.7^2 - 21.8x - 51.0y = [(21.8*51.0)xy]^1/2 And then square both sides again. This will fix your problem of dealing with the square roots of x and y, but then you have the almost as difficult problem of dealing with x^2 and y^2 in the same equation as xy. What I'd recommend trying instead is making up two new variables: u and z. Define them this way: u = x^1/2 z = y^1/2 Your first equation can be rewritten as: 69.7 = (21.8^1/2)u + (51.0^1/2)z And you can re-write your second equation similarly. Then solve for u and z just as you would for any pair of equations. Then, square u to find x and square z to find y. Finally, go back and check your answers, to make sure you didn't lose a negative or something weird like that. I hope this helps. If you have other questions about this or you're still stuck, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/
Date: 10/28/2002 at 13:46:18 From: Robert Summers Subject: Thank you (Two equations with two unknowns) As soon as you mentioned substituting u = x^1/2, it all came back to me. Thank you very much!
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