Is there another way to solve this problem? I was thinking about how to find the price of something before tax when you know the tax rate and the final price including tax, and I know that there's one way to do it, but I want to know the way to solve the problem when you're not adding like terms.
Here's the original equation I came up with:
x + x(.086) = 1.38
.086 is the tax rate and $1.38 is the final price.
I know that you can add like terms and then divide 1.38 by 1.086, but is there a way to solve the problem when you turn it into x(.086) = 1.38 - x? I know this is a slightly obscure question since there's already an established method, but I'm just wondering if there's another way where you're not combining like terms. Thanks!
 A: You can write the equation as $x=1.38-0.086x$ and use fixed point iteration.  Start with some guess at $x$, say $1$, and call it $x_0$.  Then the iteration is to write $$x_{i+1}=1.38-0.086x_i$$
and iterate to convergence.  It converges to two decimal places in three iterations and to the nine my spreadsheet displays after ten iterations.  A lot more work, but it is a useful approach for numeric answers to an equation you can't solve analytically.
A: $x + x*r = x*(1 + r)$ Always.
And the follow 4 equations are always equivalent and you can always use whichever one is more useful.  They all say the exact same thing but what concept they emphasize may differ:
$x + x*r = x*(1+r) = F$.
$x = \frac F{1+r}$ and
$x = F-x*r$
$x*r = F - x$
$x$ is original price. $F$ is final price $x*r$ is how much tax you paid. $(1+r)$ is by what proportion the final price was compared to the original price.
All of those are mutually consistant, equivalent, and basically all say the exact same things.
=== 
So if you are told
$x*.086 = 1.38 - x$ you solve it like any other algebra question.
Add $x$ to both sides to get the $x$ to one side of the equation:
$x*.086 = 1.38 - x$
$x*.086+ x = 1.38 - x + x$
$x*.086 + x =  1.38$
Then factor out the common $x$.
$x*.086 + x = 1.38$
$x(.086 + 1) = 1.38$
$1.086*x = 1.38$.
then divide both sides by $1.086$
$1.086x = 1.38$
$\frac {1.086x}{1.086} = \frac {1.38}{1.086}$
$x =\frac {1.38}{1.086} \approx 1.27$.
