Equation Solving Steps

Date: 04/28/97 at 13:58:01
From: Sarah Jett
Subject: Pre-calculus

I need to solve for x in the equation: 4-2(x-2b) = ax+3. I have tried 
several approaches, all of which seem to get me nowhere.  I surely 
would appreciate your help.

Thank you.

Date: 04/28/97 at 15:34:20
From: Doctor Wilkinson
Subject: Re: Pre-calculus

You're need to try to get x all by itself on one side of the equation.  
The most serious problem you have to begin with is that the x appears 
inside an expression in parentheses. So the first step should be to 
multiply out the expression: 2(x-2b)

This gives you: 

4 - 2x + 4b = ax + 3

Now the biggest problem you have is that x is on both sides of the 
equation. If you subtract (ax) from the right side of the equation, 
there will be no x on the right side.  But whatever you do to one side 
of an equation you must also do to the other.  So the next step is to 
subtract ax from both sides of the equation:

4 - 2x + 4b - ax = 3

Now you need to get the x terms on the left side together:

4 + 4b - 2x - ax = 3

Combine the x terms:

4 + 4b - (2 + a)x = 3

You still don't have x all by itself on the left because you have 
extra terms, which you can get rid of by subtracting them away.  But 
again, whatever you do on the left you must do on the right, so 
subtract 4 + 4b from both sides of the equation to get:

-(2 + a)x = 3 - 4 - 4b  or  -(2 + a)x = -1 - 4b

Now you've got minus signs everywhere.  Before going on, let's 
simplify by multiplying both sides by -1.   This isn't entirely 
necessary, but it does make things simpler:

(2 + a)x = 1 + 4b

Now the only remaining problem is that x is not all by itself, but is
multiplied by (2 + a).  But we can fix that easily by dividing both 
sides by (2 + a).  This gives the solution:

x = (1 + 4b)/(2 + a)

At every stage of this and similar examples, it helps to figure out 
what the worst problem you are facing is.  You are always trying to 
isolate the unknown.  In decreasing order of seriousness you have:

(a) An unknown inside parentheses.
(b) An unknown on both sides of the equation.
(c) An unknown in more than one term.
(d) An unknown multiplied by something.

Always tackle the worst problem first.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/