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### One Variable Equations with Decimals

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Date: 02/11/97 at 00:41:57
From: Charnesha
Subject: Algebra 1

I know how to solve equations, but I came across two in my math book
that I have no clue about how to solve.  I have tried doing it the way
my teacher showed me, but this type of equation is different from any
Thanks!

a)  8.25x + 3 = 17.5 + x

b)  x + 17.5 = 8.25x + 3
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Date: 02/11/97 at 02:00:54
From: Doctor Mike
Subject: Re: Algebra 1

Hello Charnesha,

One thing you probably noticed is that these 2 equations are
essentially the same. To make one look more like the other, just
switch left and right sides and reverse the addition order.

So, lets just look at the first one.  An equation involving x is just
a clue about a mystery number known only as x.  If you make the most
of that clue, you can find out what x must be.

An equation "left_side = right_side" just says that it is true that
the expressions on the 2 sides of the equal sign are the same. One of
the most important principles of high school algebra is that if you
have a true equation, and you do EXACTLY the same thing to both sides,
then what you get is also a true equation.

Let's see how this works for 8.25x + 3 = 17.5 + x.  I still do not
know what x is, but I do know it is supposed to be some number. I want
to (you will soon find out why) subtract that unknown mystery number
from both sides of the equation.  On the left I have 8.25 times x, and
3, added together.  If I subtract x from that I'll only have 7.25
times x, and I'll still have the 3, so the left side will be
7.25x + 3.  On the right I have 17.5 and x added together.  If I
subtract x from that I'll only have the 17.5 remaining, because adding
the x and immediately subtracting it off again doesn't really change
anything.  It's like adding zero.  So, the new true equation looks
like:

7.25x + 3 = 17.5

The next step is just like the first, only different!  What I mean is
that again I will be doing EXACTLY the same thing to both sides of the
equation, to get another new true equation. What's different is the
thing I do.  This time I don't subtract x from both sides, I subtract
3.  I think you can see I'll get:

7.25x = 14.5

I'm still not there yet, but I'm close.  The left side is 7.25 times
x, and if I divide the left side by 7.25 I'll get just x. So, the
final step is to divide both sides by 7.25.  Remember the basic
principle: DO EXACTLY THE SAME THING TO BOTH SIDES:

x = 14.5/7.25

You can either divide 14.5 by 7.25 by long division, or do it on a
calculator.  Or maybe you observed that 7.25 is precisely half of
14.5.  However you do it you find out the right side is 2.  So, x = 2
is a true equation that gives you the value of the mystery number x.

I hope this helps.  Have fun in Algebra 1!

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

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