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### Solve for x, y, and z

```
Date: 7/27/96 at 3:9:31
From: Anonymous
Subject: Solving System of Equations

My friend sent me these equations. I'm completely stuck! Help!

Solve the following system

1/2y + 1/3z = 26
1/3x + 1/4z = 23
1/2x + 1/4y = 28
- T.Y.
```

```
Date: 7/29/96 at 13:28:24
From: Doctor Jerry
Subject: Re: Solving System of Equations

The solution is x = 42, y = 28, and z = 36.

Your friend has given you a system of three equations in the three
unknowns x, y, and z.  Geometrically, each equation describes a plane
in space.  The solution is the one point of space common to the three
planes.

There are systematic ways of solving such systems. One is called the
Gauss/Jordan algorithm.  A less formal method that works for this
system is to multiply the first equation by 3 and the second by -4 and

3/2y + z = 3*26
-4/3x - z = -4*23

-4/3x +3/2y = -14

Now, look at this equation together with the third of the equations
you gave.

-4/3x + 3/2y = -14
1/2x + 1/4y = 28

To get rid of the fractions, multiply the first equation by 6 and the
second by 4.  This gives

-8x + 9y = -84
2x +  y = 112

Now multiply the second equation by 4 and add the two equations. This
gives

13y = 364.

So, y = 28.  From the equation 2x+y = 112, it follows that x = 42.
From the equation 1/2y + 1/3z = 26, it follows that z = 36.

again if not.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Linear Algebra
High School Linear Equations

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