GMAT Maths Algebra - Equations

Equations

  • Changing the order of the addends (numbers you're adding) doesn't change their sum (what they equal when added together).  Example: 
    •         a + (b + c) = (a + b) + c
  • Any number plus 0 (zero) equals itself.  Example: 
    •         a + 0  = a
  • If two sides of an equation are equal, you can add or subtract the same amount to both sides, and they will still be equal.  Example: 
    •         a = b
            a + c = b + c
            a - c = b - c
    Example
1.   Solve:   x + 79 = 194


     Solution:  
     x + 79 = 194
     x + 79 - 79 = 194 - 79
     x = 115		  
You need to get the variable by itself (isolate the variable).
To undo adding 79, subtract 79 from both sides.
2.   Solve:   x - 56 = 604


     Solution:  
     x - 56 = 604



     x - 56 + 56 = 604 + 56
     x = 660		  
You need to isolate the variable.
To undo subtracting 56, add 56 to both sides.
 
  • Order of operations:
    The operations inside parentheses () and brackets [] are done first. Then any operations involving exponents. Then do all multiplying and dividing from left to right.  Finally, do all addition and subtraction from left to right.

  • Multiplication can be written three different ways:

     9 * x
     9x
     9(x)


  • A fraction bar is also a division symbol.


  • Changing the order of multipliers (numbers you're multiplying together) doesn't change their product (total when the numbers are multiplied together).  Example: 

    ab = ba


  • Zero times any number is zero and 1 times any number is the number.  Examples: 

    x(0) = 0
    (0)x = 0
    x(1)= x
    1 * x = x


  • If two sides of an equation are equal, you can multiply or divide each side by the same quantity (number or equation) and it will still be equal.  Examples:

    a = b, c <> 0
    ac = bc
    (a / c) = (b / c)

    Example:
1.   Solve:   7x - 7 = 42


     Solution:  
     7x - 7 = 42
     7x - 7 + 7 = 42 + 7
     7x = 49
     (7x) / 7 = 49 / 7
     x = 7		   The variable needs to be isolated.
To undo subtracting 7, add 7 to both sides.
Adding 7 hasn't isolated the variable, so we need to continue.
To undo multiplying by 7, divide both sides by 7.
2.   Solve:   5(x + 2) = 25


     Solution:  
     5(x + 2) = 25
     [5(x + 2]/5 = 25/5
     x + 2 = 5
     x + 2 -2 = 5 -2
     x = 3		  
The variable needs to be isolated. To undo
multiplying by 5, divide by 5 on both sides.
Dividing by 5 hasn't isolated the variable, so we need to continue.
To undo adding 2, subtract 2, from both sides.