Equations

Combining Like Terms

 

In an algebra expression, symbols, called variables, are used to represent numbers.  In many cases, that number is a value that we don’t know and the problem is asking us to find what it is.  Algebraic expressions often look like this:  3x – 1.  In this expression, a number that we don’t know (x) is being multiplied by 3, then 1 is being subtracted from that. 

 

A ‘term’ in algebra refers to any number and any variable that it is being multiplied by.  In the expression 3x – 1, 3x is a term (it includes a coefficient, 3, and a variable, x) and -1 is a term.  In an algebra expression, terms that are multiplied by the same variable, or no variable, are called like terms, and they can be combined using addition or subtraction.

 

 If you have several terms in an expression like this:  5x + 3 – 2x + 7, the like terms can be combined to make a simpler expression.  5x – 2x combines to make 3x, and 3 + 7 combined to make 10, so the new expression is 3x + 10. 

 

Different variables and like variables raised to different powers are not like terms.  In the expression 5x + 3y – x2 + 9, there are no like terms, and this expression is as simple as it can be.

 

Distribution

 

Sometimes, an expression may have parentheses that can add another step to combining like terms.  If you have a number or variable times an expression with a variable in parentheses, you can distribute the number by multiplying it times each of the values in the parentheses.  For example, if you have 3(x – 5), you can multiply the 3 times each of the items in parentheses.  3 times x can be written as 3x and 3 times -5 is -15.  That simplifies the expression to 3x – 15.

 

Solving for a Variable

 

In order to find a missing variable, it needs to be part of an equation.  An equation is similar to an expression, but it contains an equals sign.  The equals sign is important because it gives us a key piece of information:  whatever numbers and variables are on the left side of the equals sign must be equal to the numbers and variables on the right side.  By moving numbers around and combining like terms, we can simplify an equation down to having only the variable on one side of the equals sign and a number on the other, which reveals what the variable is.

 

If you have a complicated equation, use this order to solve for the variable:

 

1. Combining like terms (this includes distribution and making sure all of the terms with the variable are on the same side of the equals sign).

 

2. Addition and subtraction operations

 

3. Multiplication and division operations

 

Addition and Subtraction Operations

 

After combining like terms, your next step in solving for a variable is dealing with any addition or subtraction in the equation.  This is done by performing the inverse, or opposite operation.  Say you have the equation x + 3 = 5.  Think of the equation as a scale balance.  Both sides must be equal, or the statement is not true.  Therefore we can add or subtract something from either side, as long as we do the same to the other side.  If you want to get x alone, and the expression on one side is x + 3, you can get x alone by performing the inverse, or opposite, operation.  The inverse of x + 3 is to subtract 3, which would leave only x on the left side, but if you subtract 3 from the left side, you must do the same to the right side and subtract 3 from 5. 

 

x + 3 = 5

    -3       -3

x = 2

 

Multiplication and Division Operations

 

Similar to addition and subtraction, multiplication and division problems can be solved by performing the inverse operation.  If a number is multiplied by the variable, the inverse operation is to divide by that number.  You must then remember to divide the other side of the equation by the same number.  Likewise, if the variable is divided by a number, the inverse operation is to multiply by that number, which you must also multiply the other side of the equation by.

 

Multiplication:

 

3x = 15

÷3    ÷3

x = 5

 

Division:

 

x ÷ 2 = 5

•2         •2

x = 10

 

Example:

 

2(3x + 1) = 20

First, distribute and combine like terms if there are any

6x + 2 = 20

Then, addition and subtraction operations

6x + 2 = 20

-2           -2

6x = 18

Then multiplication and division operations

6x = 18

÷6     ÷6

x = 3