Associated Topics || Dr. Math Home || Search Dr. Math

### Basic Tips on Solving for X

```
Date: 11/16/2000 at 22:25:13
From: Megan
Subject: How to solve for the variable "X"

The following problems have tripped me up. I can't seem to figure out
how to solve them. I've tried, several times, and had my algebra
teacher explain them, but I still can't get the answers.

3x - 11 = 6
2x - 7  = 3 + x
3 + 3x  = 9 + 2x
3x - 3  = 6 + x

Thanks,
Megan
```

```
Date: 11/17/2000 at 11:03:31
From: Doctor Ian
Subject: Re: How to solve for the variable "X"

Hi Megan,

x = 5

That's no fun, is it? There's nothing to figure out. But I can
multiply both sides of the equation by 3 without changing the truth of
the equation, right?

3x = 15

And there are a lot of different ways I can write 3:

3 = 1 + 2

= 7 - 4

= -5 + 8

So I can change my equation to look like this:

3x = 15

x(3) = 15

x(7 - 4) = 15

7x - 4x = 15

And since I can add or subtract anything on both sides of an equation
without changing the truth of the equation, I can add 4x to both
sides:

7x - 4x = 15

7x - 4x + 4x = 15 + 4x

7x = 15 + 4x

And of course, I can do the same thing with any number I want:

7x = 15 + 4x

7x - 5 = 15 + 4x - 5

7x - 5 = 15 - 5 + 4x

7x - 5 = 10 + 4x

Now I have something that looks just like your problems, don't I?

The trick, then, is to reverse the steps that I just did, to get back
to the original, simple, boring equation:

7x - 5 = 10 + 4x

7x - 5 + 5 = 10 + 4x + 5           Add 5 to both sides

7x = 15 + 4x

7x - 4x = 15 + 4x - 4x          Subtract 4x from both sides

3x = 15

(3x)/3 = 15/3                  Divide both sides by 3

x = 5

It's easy to forget what's really going on when you're solving an
algebraic expression for a variable. You're really just trying to
rewrite a complicated expression in a simpler way, without doing
anything to change the meaning of the expression. It's a little like
what you would do to change a complicated sentence like:

"Under a mammal of feline persuasion is located, at this time,
in an unspecified location, a rectangle of woven fabric."

into a simpler one like:

"The cat is on the mat."

Algebra is basically tying knots and then untying them again.

With that in mind, let's look at one of your example problems:

3 + 3x = 9 + 2x

The first thing to notice is that we have numbers on both sides. We
know that someone put them there by adding a number to both sides of
the equation, so we can un-put them there by subtracting a number from
both sides:

3 + 3x = 9 + 2x

3 + 3x - 3 = 9 + 2x - 3

(3 - 3) + 3x = (9 - 3) + 2x

3x = 6 + 2x

Now we have multiples of x on both sides. We can do the same kind of
thing to fix that:

3x = 6 + 2x

3x - 2x = 6 + 2x - 2x

(3 - 2)x = 6 + (2 - 2)x

x = 6

Can you follow the same kinds of steps to solve the rest of the
problems?

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search