Page 178 - ArithBook5thEd ~ BCC

6.4 Linear terms

A linear term is an expression of the form

Cx or Dy or Ez

where C (or D or E, etc.) is a nonzero number called the coefficient,and x (or y or z,etc.) is a
variable. (You will note that linear terms show up in linear functions!) For example,

1
3x, −5y, z, w, −t
2
1
are linear terms with coefficients 3, −5, 1, ,and −1, respectively. Note that a variable by itself
2
(without a coefficient) is understood to have a coefficient of 1 (not 0). Also, a negative sign is sufficient
to indicate the coefficient −1.
Linear terms with the same variable can be combined into a single term by addition and subtraction.
For example

5x +4x =(5 + 4)x =9x
3x +2x − x +4x =(3 + 2 − 1+ 4)x =8x

t +5t − 8t =(1 + 5 − 8)t = −2t
2 " 2 # 4
−2y + y = −2+ y = − y.
3 3 3
We are simply using the distributive law here. This is possible because, in each expression, the value
of the variable, though unspecified, is the same wherever it occurs. Note that it is not possible to
combine linear terms with different variables. Thus, for example, the expressions

5x +3y or z − 3t

cannot be simplified in any way. This is because x and y (or z and t) are independent variables – they
need not take on the same value.
Linear terms with the same variable are called like terms.
Example 224. Simplify the following expression by combining like terms.

5x − 4y +3y − 11x +2t.

Solution. First survey the expression to determine the different types of like terms: there are x-terms,
y-terms, and t terms. There is just one t term, so that term won’t combine with any other term.
There are two x-terms, and two y-terms. The following approach is convenient because it allowsus to
rearrange the expression so that like terms are next to eachother.

Treat subtractions as additions of negative terms.

(Recall that subtraction means ‘adding the opposite.’) Doing this, our expression becomes

5x +(−4y)+ 3y +(−11x)+ 2t.

Page 178