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A L G E B R A

8

The rule for subtraction

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 6.)

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

x² − 2xy + y²

What number is the coefficient of x²?

1

What number is the coefficient of xy?   −2

What number is the coefficient of y²?    1

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.

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.

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.  Lesson 6.

Problem 2.   What number is the coefficient of x?

 a) x2 . 12 . Compare Lesson 5, Problem 7b.
 b) 3x 4 . 34 . 3x 4 = 34 · x Lesson 4.

Problem 3.   How do we add like terms?

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

 a)  6x + 2x = 8x b)  6x − 2x = 4x c)  5x + x = 6x d)  5x − x = 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)  −x − x = −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

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 6.    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 + 2c − d = −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)   (x − y) − (y + xy − x) − (2x − 4xy − 2y) = x − y −y − xy + 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 7.   5abc + 2cba.  Are these 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.

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

c)   9xyz + 3yzx + 5zxy  = 17xyz

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

a)   2n + 2 − n =  n + 2

b)   n − 2 − 3n + 1 =  −2n − 1

c)   2n + 4 − 2n − 2 = 2

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.

Problem 10.   Subtract  4a − 2b  from  a + 3b.

Change the signs of the subtrahend, and add:

a + 3b − 4a + 2b = −3a + 5b.

Problem 11.   Subtract  x² − 5x + 7  from  3x² − 8x − 2.

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

Next Lesson:  Linear equations

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