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8

ADDING LIKE TERMS

WHEN NUMBERS ARE ADDED OR SUBTRACTED, we call them terms. (Lesson 1.)  Like terms look exactly alike, except perhaps for a numerical factor, which is called the coefficient of the term.

Here is a sum of like terms:

4x² − 5x² + x²

Each term has the same literal factor, x²; only the coefficients are different.  The coefficient of x² in the first term is 4.  The coefficient in
the second term is −5.  We include the minus sign. See Naming terms in Lesson 3.  And in the last term, the coefficient of x² is understood to be 1. For, x² = 1x². (Lesson 5.)

Here, on the other hand, is a sum of unlike terms:

x² − 2xy + y²

What number is the coefficient of x²?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

 1 

What number is the coefficient of xy?   −2

What number is the coefficient of y²?    1  

Adding like terms

In this sum --

2x + 3y + 4x − 5y

-- the like terms are 2x and 4x,  3y and −5y.

What do we do with like terms?  We add, or combine, them:

2x + 3y + 4x − 5y = 6x − 2y.

That is, we add (Lesson 3) their coefficients.  The order of the terms does not matter.

We say that there are two terms "in" x and two "in" y. The preposition "in" indicates which terms are the like terms.

See Problem 12.

Problem 1 .   6x − 4yz

a)  What number is the coefficient of x  6 

b)  What number is the coefficient of y ?  −4  

c)  What number is the coefficient of z ?   −1.  −z = (−1)z.

See Lesson 5.

Actually, the coefficient of any factor is all the remaining factors. Thus in the term 4ab, the coefficient of a is 4b; the coefficient of 4a is b; and so on. In this term, x(x − 1), the coefficient of (x − 1) is x.

Problem 2.   In the expression 5ayx, name the coefficient of

a)  x  5ay          b)  y  5ax          c)  yx  5a 

d)  5a  xy          e)  5  ayx 

Problem 3.   In this product  2(x + y)z

a)  name each factor.   2, (x + y), z

b)  name the coefficient of z.   2(x + y)

c)  name the coefficient of (x + y).   2z

Problem 4.   What number is the coefficient of x?

  a)   x
2
   1
2
  Compare Lesson 6, Problem 7b.
  b)   3x
 4
   3
4
   3x
 4
3
4
· x    Lesson 4.

Problem 5.   How do we add like terms?

Add their coefficients; make that sum the coefficient of the common factor.

Problem 6.   Add like terms.

   a)  6x + 2x = 8x   b)  6x − 2x = 4x
 
   c)  5x + x = 6x   d)  5xx = 4x
 
   e)  −4x + 5x = x   f)  4x − 5x = −x
 
        It is the style in algebra not to write the coefficients 1 or −1.
 
   g)  −5x − 3x = −8x   h)  −xx = −2x

i)  −3x − 4 + 2x + 6  = −x + 2

j)  x − 2 − 4x − 5  = −3x − 7

k)  4x + y − 2x + y = 2x + 2y

l)  3xy − 8x + 2y  = −5x + y

m)  4x² − 5x² + x² = 0 

Problem 7.   Add like terms.

a)   2a + 3b   These are not like terms. The literals are different.

b)   2a + 3b + 4a − 5ab  = 6a + 3b − 5ab.
     Terms that you cannot combine, simply rewrite.

Problem 8.    Remove parentheses and add like terms.

  a)   (2a − 3b + c) + (5a − 6b + c)   =   2a − 3b + c + 5a − 6b + c
 
    =   7a − 9b + 2c
  b)    (a + 2b + 4c − 3d) − (3a − 8b − 2c + d)
 
   = a + 2b + 4c − 3d −3a + 8b + 2cd
 
   = −2a + 10b + 6c − 4d
  c)    (4x − 3y) + (3y − 5x) + (5z − 4x)
 
   = 4x − 3y + 3y − 5x + 5z − 4x
 
   = −5x + 5z
  d)    (5xy − 3x + 2y − 1) − (2xy − 7x − 8y + 6)
 
   = 5xy − 3x + 2y − 1 −2xy + 7x + 8y − 6
 
   = 3xy + 4x + 10y − 7
  e)   (xy) − (y + xyx) − (2x − 4xy − 2y)
 
   = xyyxy + x − 2x + 4xy + 2y
 
   = 3xy
  f)   (4x² − 7x − 3) − (x² − 4x + 1)
 
   = 4x² − 7x − 3 − x² + 4x − 1
 
   = 3x² − 3x − 4
  g)   (6x3 + 4x² − 2x − 6) − (2x3 − 8x² + x − 2)
 
   = 6x3 + 4x² − 2x − 6 − 2x3 + 8x² − x + 2
 
   = 4x3 + 12x² − 3x − 4
  h)   (x² + x + 1) + (2x² + 2x + 2) − (x² − x − 1)
 
   = x² + x + 1 + 2x² + 2x + 2 − x² + x + 1
 
   = 2x² + 4x + 4

Problem 9.   5abc + 2cba.  Are those like terms?

Yes.  The order of factors does not matter.
Upon adding those like terms, we get 7abc.
When writing the final answer, it is conventional to preserve the alphabetical order.

Problem 10.   Add like terms.

   a)    4xy − 9yx  = −5xy   b)    8x − 5xy − 4x + 4yx  = 4xxy

c)   9xyz + 3yzx + 5zxy  = 17xyz

d)   3xy − 4xyz + 3x − 8yx + 5yzx − 9x  = −5xy + xyz − 6x

Problem 11.   Add like terms.

a)   2n + 2 − n =  n + 2

b)   n − 2 − 3n + 1 =  −2n − 1

c)   2n + 4 − 2n − 2 = 2

Problem 12.   Add like terms, which are in (x + 2). Do not remove parentheses.

a)  3(x + 2) + 7(x + 2) =  10(x + 2).

b)  2(x + 2) − 5(x + 2) =  −3(x + 2).

c)  x(x + 2) + 4(x + 2) =  (x + 4)(x + 2).

We added the coefficients.

d)  x(x + 2) − (x + 2) =  (x − 1)(x + 2).

Problem 13.   Add like terms, which are in x or y. Add the coefficients.

a)   px + qx = (p + q)x.

b)   ax + bycx + dy = (ac)x + (b +d)y.

c)   x + ax = (1 + a)x.      d)   axx = (a − 1)x.

e)   (a + b)x + cx = (a + b + c)x.

f)   (ab)xcx = (abc)x.

f)   (a + b)x − (b + a)x = 0.

Problem 14.   Add like terms.

a)  3a2b3 − 2ab2 + a3b2 − 5b2a + b3a2 = 4a2b3 − 7ab2 + a3b2.

b)   xy2xy + x2yy2x + 2yx2 + yx = 3x2y.

*

In calculus, the student will not see any problem in the form "Subtract a from b." However, in certain standard exams that form tends to come up. Hence, the following rule.

The rule for subtraction

"Subtract a from b."  Is that  ab  or  ba ?

It is  ba.   a is the number being subtracted.  It is called the subtrahend.  The subtrahend appears to the right of the minus sign -- before the word "from."

Example.   Subtract  2x − 3  from  5x − 4

 Solution.   2x − 3 is the subtrahend.

(5x − 4) − (2x − 3) = 5x − 4 − 2x + 3
 
  = 3x − 1.

  Notice:  The signs of the subtrahend change.

2x − 3  changes to  −2x + 3.

We can therefore state the following rule for subtraction.

Change the signs of all the terms in the subtrahend.
Then add the like terms.

Problem 15.   Subtract  4a − 2b  from  a + 3b.

Change the signs of the subtrahend, and add:

a + 3b − 4a + 2b = −3a + 5b.

Problem 16.   Subtract  x² − 5x + 7  from  3x² − 8x − 2.

  3x² − 8x − 2 − x² + 5x − 7
 
2x² − 3x − 9.
end

Next Lesson:  Linear equations

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