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### Solving Systems of Equations

```
Date: 10/30/96 at 11:52:48
From: sylvia hobson
Subject: algebra

I thought I knew how to do this, but guess I don't.
Solve for a, b, and c in at least two ways:

2a+6b+c = 5
a+b-c = -1
a+8b+2c = 0

```

```
Date: 10/31/96 at 17:40:57
From: Doctor Tracy
Subject: Re: algebra

Sylvia,

You are right that there are many approaches to solving this problem.
They are really all equivalent mathematically, but everybody seems to
prefer one approach or another.  I will show you how to solve your
problem using two techniques and then you should be able solve other
problems like this one.

First of all, let's label your equations:

(I)      2a + 6b + c = 5
(II)       a +  b - c =-1
(III)       a + 8b +2c = 0

Technique 1:

The first thing that we want to do is eliminate one of the variables.
Let's try to eliminate a.  Notice that if you add -2 times the second
equation to the first equation, the a's will cancel:

(I)
2a + 6b +  c =  5
-2*(II) -2a - 2b + 2c =  2
----------------------------
0 + 4b + 3c =  7

Do this kind of thing to eliminate a again.  I notice that equation
(II) minus equation (III) will eliminate a and give me:

0 - 7b - 3c = -1 ------> 7b + 3c = 1

Now we want to use our two new equations to eliminate either b or c.
Since our first equation minus the second one will get rid of c,
that's what I'll do. (If for some reason you really wanted to get rid
of b at this point, you could multiply the first equation by 7 and the
second equation by -4 and add.)

Eliminating c we get -3b = 6, so b = -2.  Use this in one of the
equations with only b and c to find c and then use your b and c in one
of the original 3 equations to find a.

Technique 2:

First take one of the equations and solve for one of the variables:

I will take equation (II) and solve for a to get: a = -1 - b + c.

Now, take one of the other equations and solve for the same variable:

I will take equation (III) and solve for a to get: a = -8b - 2c.

Put these two together to get: -1 - b + c = -8b - 2c.

Now collect like terms to get: -1 + 7b + 3c = 0

Solve this equation for one variable: b = (1-3c)/7.

Replace b by this expression in one of the two equations for a above:
i.e., take a = -1 - b + c and replace b to get:

a = -1-[(1-3c)/7]+c -----> a = (6+10c)/7

Finally, go back to one of your original equations (I'll use (II)) and
replace a and b with the new expressions in c.  Take the equation
a+b-c = -1 and replace a and b to get:

(6+10c)/7 + (1-3c)/7 - c = -1.

Now, this is an equation in only c, so you should be able to solve it
for c.  Once you have c, plug it in to the equations a = (6+10c)/7 and
b = (1-3c)/7 to find a and b.

I hope this helps you.

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

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