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### Simplifying Mixed Algebraic Expressions

```
Date: 05/08/98 at 20:07:12
From: Erin
Subject: Simplyfing mixed expressions(algebra)

I have a test on Monday, and I can't get the answers to these

x + y
2x - ----- = ?
y

8
------ - 6 = ?
3a - 1

1
(x - 4) - ----- = ?
x - 2

x
-- - (x + 2) = ?
2y

7
(a + 2) + ----- = ?
a - 2

3       1
4 - ----- - ----- = ?
y - 1   y + 1

```

```
Date: 05/12/98 at 14:42:15
From: Doctor Jeremiah
Subject: Re: Simplyfing mixed expressions(algebra)

To simplify an expression where part is divided by one value and part
is divided by another value, you need to find common denominators to
make both be divided by the same value.

You know that you can modify a piece of an expression by multiplying
it by 1:

x + y          y   x + y
2x - ----- = (2x) * - - ----- =
y            y     y

See how we multiplied 2x by y/y (which is exactly 1)? Thus:

x + y          y   x + y   2xy   x + y   2xy - x - y
2x - ----- = (2x) * - - ----- = --- - ----- = -----------
y            y     y      y      y         y

We need to do the same thing with the next problem:

8              8          (3a - 1)
------ - 6 = -------- - 6 * --------
3a - 1       (3a - 1)       (3a - 1)

8 - 6(3a - 1)   9 - 18a - 6   3 - 18a
= ------------- = ----------- = -------
3a - 1         3a - 1     3a - 1

It's always the same. All you need to do is to match the denominator
for all the pieces.

The last one you give is the most interesting one:

3       1
4 - ----- - -----
y - 1   y + 1

Here, we have three parts. Obviously, getting the denominator to be
(y + 1) or even (y - 1) is not good enough. The only way to get them
all to match is to use the product of all the denominators:
(y - 1)(y + 1). Thus:

3       1     4(y - 1)(y + 1)      3(y + 1)        1(y - 1)
4 - ----- - ----- = --------------- - -------------- - --------------
y - 1   y + 1    (y - 1)(y + 1)   (y - 1)(y + 1)   (y - 1)(y + 1)

4(y - 1)(y + 1) - 3(y + 1) - (y - 1)   4y^2 - 4y - 8
= ------------------------------------ = -------------
(y - 1)(y + 1)                   y^2 - 1

Does that make any sense?

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

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