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Solve 2x + 3 = 10

Date: 09/17/2001 at 20:43:24
From: Billy
Subject: Something like 2x+3=10

My problem: 2x + 3 = 10

You are supposed to subtract 3 from the 10, which is 7, then divide by 
two, which is 3.5x + 10. I don't get what to do next to simplify the 

Date: 09/18/2001 at 10:21:40
From: Doctor Rick
Subject: Re: Something like 2x+3=10

Hi, Billy.

The goal is to *solve* the equation; that is, to find the value of x 
for which the equation is true. There are two basic principles that we 
use to do this. 

One I call the principle of "undoing." Look at the expression on the 
left, 2x+3. According to the order of operations, it is built out of 
just-plain-x in two steps: first multiply the x by 2, then add 3. We 
want to get just-plain-x alone on one side of the equation. To do 
this, we "undo" the steps by which the expression was built.

When you get dressed, first you put on your socks, then you put on 
your shoes. When you get undressed, first you take off your shoes (the 
*last* thing you put on), then you take off your socks. So here: we 
undo the last thing first. That's why you first subtract 3, which 
undoes the addition of 4. Then you divide by 2, which undoes the 
multiplication by 2.

The other principle is "keeping the balance." The equation is like a 
scale with two pans that balance. They balance because the two pans 
contain the same weight. In the same way, the two sides of the 
equation balance because have the same value - they equal the same 
number. (We don't know what that number is yet.)

We want to *keep* the equation balanced. If we take a weight off one 
pan of a balance scale, we must take the same amount off the other 
side, or it will no longer be balanced. In the same way, if we do 
something to one side of the equation, we must do the same to the 
other side, otherwise it will go out of balance. So, when we subtract 
3 from one side, we must subtract 3 from the other side, too. When we 
divide one side by 2, we must divide the other side by 2 also.

Now I'll show you how these principles work to solve your equation.

  2x + 3 = 10

First we subtract 3 from *both* sides. Then we have

  2x = 10 - 3

Doing the subtraction on the right, we have

  2x = 7

Then we divide *both* sides by 2. Then we have

  x = 7/2

That's the solution: if x is 7/2 then the equation is true. Let's 
check this out: I will replace x by 7/2 in the equation.

  2(7/2) + 3 = 10

  7 + 3 = 10

  10 = 10

Both sides have the same value, so the equation balances. If we chose 
a different value for x, then the final equation, x = 7/2, wouldn't 
balance, and so the original equation, 2x + 3 = 10, wouldn't balance, 
either: 2x + 3 would not be equal to 10.

- Doctor Rick, The Math Forum   
Associated Topics:
Middle School Algebra

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