Pre-Algebra: Basic Equations

This page is designed to help you better understand, work with, and solve equations. Click any of the links below to go to that section and begin understanding equations.

Addition and/or subtraction in equations.
Multiplication and/or division in equations.
Combinations of the basic operations in equations.
Quiz on Basic Equations.

Addition & Subtraction

Equations are something that you will constantly be using throughout your math career. Learning and understanding the basics is an integral part of "getting off on the right foot" when dealing with math.

This section will help you better understand, work with, and solve equations when they have addition and/or subtraction in them.

Important Things to Remember

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     
```

The Tutorial
When solving equations, remember that addition and subtraction are inverse operations -- they undo each other (i.e., 10 + 9 - 9 = 10). To solve equations using addition and subtracting, first decide which operation has been applied, then use the inverse operation to undo this (remember to add or subtract from both sides of the equation).

Examples

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

2.  Solve:    x - 56 = 604
    Solution:
    x - 56 = 604               You need to isolate the variable.
    x - 56 + 56 = 604 + 56     To undo subtracting 56, add
                               56 to both sides.
    x = 660

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Multiplication & Division

This section will help you understand, work with, and solve equations of a slightly more complex nature -- equations invloving the use of multiplication and/or division.

Important Things to Remember

Order of operations:
The operations inside of parentheses () and brackets [] are done first.
Then any operations involving exponents (which you will learn about later).
Then do all multiplying and dividing from left and 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 (one) 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)     
```

The Tutorial
When solving equations, remember that multiplication and division are inverse operations, therefore they undo each other (i.e., (4 * 8)/8 = 4). To solve equations using multiplication or division, first decide which operation has been applied, then use the inverse operation to undo this (remember to multiply or divide on both sides of the equation).

Examples

1.  Solve:    6x = 36
    Solution:
    6x = 36                    You need to get the variable by itself
                               (isolate the variable).
    (6x)/6 = 36/6              To undo multiplying by 6, divide by
                               6 on both sides.
    x = 6                      

2.  Solve:   x / 5 = 10
    Solution:
    x / 5 = 10                 You need to isolate the variable.
    5(x / 5) = 10(5)           To undo dividing by 5, multiply both
                               sides by 5.
    x = 50

Back to the top.

Combination of Operations

This section will help you understand, work with, and solve complex equations that involve different combinations of multiplication, division, addition, and subtraction.

Important Things to Remember

Order of operations:
The operations inside of parentheses () and brackets [] are done first.
Then any operations involving exponents (which you will learn about later).
Then do all multiplying and dividing from left and right.
Finally, do all addition and subtraction from left to right.
Multiplication can be written three different ways:
```
     7 * x
     7x
     7(x)     
```
A fraction bar is also a divison symbol.
Also, be sure to refer to the above sections if you have forgotten or need to review any of the other material covered.

The Tutorial
When solving complex equations, like the ones used in the examples below, be sure to remember that multiplication and division are inverse operations along with addition and subtraction. Therefore, they undo each other (i.e., (5 * 2)/2 = 5 or 10 + 4 - 4 = 10). To solve these equations, first decide which operation has been applied and then use the inverse operation to undo this (remember to apply the operation to both sides of the equation).

Examples

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

2. Solve:   5(x + 2) = 25
   Solution: 
   5(x + 2) = 35              The variable needs to be isolated.
   [5(x + 2)]/5 = 25/5        To undo multiplying by 5, divide by 5 on
                              both sides.
   x + 2 = 5                  Multiplying by 5 hasn't isolated the
                              variable, so we need to continue.
   x + 2 - 2 = 5 - 2          To undo adding 2, subtract 2 from both
                              sides.
   x = 3

Back to the top.

Take the quiz on Pre-Algebra equations. (Very useful to review or to see if you've really got this topic down.) Do it!

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