Solving Basic Equations with Variables

Date: 11/28/2004 at 12:05:05
From: Liz
Subject: I don't understand this question and I have a test soon.

How do I solve an equation like 8x + 6 = 3x + 21?  I don't know how to
start it.

Date: 11/29/2004 at 08:43:35
From: Doctor Boyd
Subject: Re: I don't understand this question and I have a test soon.

Hi Liz,

Thanks for writing in to Dr. Math.

Let's look at a similar question:

  3x + 1 = 2x + 4

We could think about this graphically, with a rectangle being an x and
a square being a 1:
   _____   _____   _____   _     _____   _____   _   _   _   _
  |_____| |_____| |_____| |_| = |_____| |_____| |_| |_| |_| |_|

The equal sign means that the left side is balanced by the right side.
Just like on a balance scale, if I take the same amount away from each
side (or add the same amount to each side) the scale will stay
balanced.  So I notice I can take away two of the x's from each side,
which leaves me with:
   _____   _     _   _   _   _
  |_____| |_| = |_| |_| |_| |_|

Also, I can take one of the 1's away from each side:
   _____     _   _   _
  |_____| = |_| |_| |_|

So x = 3.

This can also be done with algebra:

         3x + 1 = 2x + 4

    3x + 1 - 2x = 2x + 4 - 2x    -->(Subtract 2x from each side)

    3x - 2x + 1 = 2x - 2x + 4    -->(Collect like terms)

         1x + 1 = 4              -->(Simplify)

     1x + 1 - 1 = 4 - 1          -->(Subtract 1 from each side)

             1x = 3              -->(Simplify)

              x = 3

Has this helped you with your question?  Please write back if you'd
like some more help with this, or if you have any other questions.

All the best,

- Doctor Boyd, The Math Forum
  http://mathforum.org/dr.math/