How to Solve Equations

Solving an equation means getting the variable (we'll use x) on one side of the equal sign all by itself. So if you have the equation
ax + b = c
then solving it means getting the x on one side of the equal sign by itself.
("a","b"', and "c" stand for real numbers).

You can perform the same operation (add, subtract, multiply, divide) on both sides of the equation. So you could add 3 to both sides like this:

ax + b     = c
       + 3     + 3
-------------------
ax + b + 3 = c + 3
This equation is true also, just like the first. But it didn't do you any good! The idea is to get rid of terms on the same side of the equation as x. Look carefully at the equation to see what operation you can "undo". Then choose the opposite. So you would like to get rid of the "+b", by subtracting:
   ax + b  = c
      - b  = c - b 
--------------------
   ax      = c - b
Notice that the +b -b leaves nothing. Now, since x is multiplied by a, you divide both sides of the equation by a:
    ax    c - b 
  ---- = -----
    a      a 
So the a's cancel and we have:
    c - b
x = -----
      a
The general rules for doing this are:
  1. Always do an operation to BOTH sides of the equation
  2. Do adds and subtracts first, then multiplies and divides
  3. Always do the opposite operation to move the term to the other side of the equation:
    • If a term is added to x, then subtract it from both sides
      Example: x + 5 = 15 ----> you subtract 5 from both sides
    • If a term is multiplied by x, then divide both sides by it
      Example: 3x = 15 ----> you divide both sides by 3
    • If x is divided by a term, then multiply both sides by it
      Example: x/2 = 20 ---> you multiply both sides by 2
Example 1:

3x + 5 = 17   <---- The original equation

3x + 5 = 17
   - 5   -5   <---- Step 1: subtract 5 from both sides
------------                 to remove it
3x     = 12

3x     = 12
---     ---   <---- Step 2: divide both sides by 3   
3         3                 to remove the 3

x      = 4    <---- The answer!!!
Example 2:

x/3 - 2   =   3  <---- The original equation


x/3 - 2   =   3
    + 2      +2  <---- Step 1: add 2 to both sides
----------------               to remove the -2
 x/3      =   5       


  
3x/3      =  3 x 5 <---- Step 2: multiply both sides
                                 by 3 to remove the 3

 x        =    15  <---- The answer!!! (3's cancelled)
Example 3:

 35  -  3x =  14 <---- The original equation


 35  -  3x =  14
-35          -35 <---- Step 1: subtract 35 from 
------------------             both sides to remove it
       -3x = -21


       -3x = -21
      ----   ----
       -3    -3  <---- Step 2: divide both sides by -3         
                                to remove the -3

         x =  7  <---- The answer!!! 
                   (remember, a negative divided
                    by a negative is a positive)