Solving Equations

The basic principle in solving equations is to get all variable terms on one side of the equal sign and all non variable terms on the other side of the equal sign. This is generally accomplished by doing the opposite operation to a term to move it to the other side. (e.g. if a term is added, you would subtract it to move it to the other side.) See examples below of different type of equation solving.

  1. Basic equations:
x + 3
=
 7  
2x
=
 14
x + 3 - 3
=
 7 - 3  
2x
=
 14
x
=
 4  
2
   2
       
x
=
 7
  1. Two step equations:
2x - 5
=
 11
2x - 5 + 5
=
 11 + 5
2x
=
 16
2
=
 2
x
=
 8
  1. Multi-step equations: First simplify each side of the equation, then move variable terms to one side of the equal sign and all non-variable terms to the other side of the equal sign.
3x + 5
=
 2x - 7  
4(x - 3)
=
 2(3x -4)
3x + 5 - 5
=
 2x - 7 - 5  
4x - 12
=
 6x - 8
3x
=
 2x - 12  
4x - 12 + 12
=
 6x - 8 + 12
3x- 2x
=
 2x - 2x - 12  
4x
=
 6x + 4
x
=
 -12  
4x - 6x
=
 6x -6x + 4
   
-2x
=
 4
       
-2
=
 -2
   
x
=
 -2
2x + 3(2x - 4)
=
 4(3x - 5) + 4        
2x + 6x - 12
=
 12x - 20 + 4        
8x - 12
=
 12x - 16        
8x - 12 + 12
=
 12x - 16 + 12        
8x
=
 12x - 4        
8x - 12x
=
 12x - 12x - 4        
-4x
=
 -4        
-4
=
 -4        
x
=
 1        
  1. Equations with fractional coefficients: First mulitiple all terms of the equation by the lowest common multiple. Then follow the same steps above.
=
  
=
8x + 9
=
 10x - 12  
2(x + 3)
=
 5(2x - 1) + (2)(10)
8x + 9 - 9
=
 10x - 12 - 9  
2x + 6
=
 10x - 5 + 20
8x
=
 10x - 21  
2x + 6
=
 10x + 15
8x - 10x
=
 10x - 10x - 21  
2x + 6 - 6
=
 10x + 15 - 6
2x
=
 -21  
2x
=
 10x + 9
2
=
   2  
2x - 10 x
=
 10x - 10x + 9
x
=
 -21  
-8x
=
   9
  2  
-8
=
 -8
       
x
=
   -9
   
  8
  1. Quadratic Equations: Follow these three steps:
    1. Set equation equal to 0
    2. Factor
    3. Solve (remember you will usually have 2 or more solutions!)
x2 + 9x
=
 -20  
x2 -4x - 6
=
 -7x + 12
x2 + 9x + 20
=
 0  
x2 - 4x + 7x - 6 - 12
=
 0
(x + 5)(x + 4)
=
 0  
x2 + 3x - 18
=
 0
x
=
 -5 or -4  
(x + 6)(x - 3)
=
 0
   
x
=
 -6 or 3
  1. Equations with rational expressions: Multiply all terms of the equation by the lowest common denominator to eliminate the fractions. Then solve as above recognizing when you have a quadratic equation and when you don't!
=
  
=
=
  
=
x2
=
 -2x + 15  
6(x + 2)
=
 21 +1(x + 2)(x -2)
x2 + 2x - 15
=
 0  
6x + 12
=
 21 + x2 -4
(x + 5)(x - 3)
=
 0  
x2- 6x - 12 + 21 - 4
=
 0
x
=
 -5 or 3  
x2 - 6x +5
=
 0
   
(x - 1)(x - 5)
=
 0
   
x
=
 1 or 5

 

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