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### 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!
```
Associated Topics:
High School Basic Algebra
High School Exponents
High School Square & Cube Roots

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