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A L G E B R A

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OF

# ALGEBRA

The rule of symmetry

ALGEBRA, we can say, is a body of formal rules. They are rules that show how something written one form may be rewritten in another form.  For what is a calculation if not replacing one set of symbols into another?  In arithmetic we replace  '2 + 2'  with  '4.'  In algebra, we may replace  'a + (−b)'  with  'a − b.'

a + (−b) = ab.

We call that a formal rule. The = sign means  "may be rewritten as"  or  "may be replaced by."

Here are some of the basic rules of algebra:

 1· a = a. (1 times any number does not change it. Therefore 1 is called the identity of multiplication.) (−1)a = −a. −(−a) = a. (Lesson 2) a + (−b) = a − b. (Lesson 3) a − (−b) = a + b. (Lesson 3)

Associated with these -- and with any rule -- is the rule of symmetry:

If  a  =  b,  then  b  =  a.

For one thing, this means that a rule of algebra goes both ways.

Since we may write

 p + (−q) = p − q -- that is, in a calculation we may replace  p + (−q)  with  p − q -- then, symmetrically: p − q = p + (−q).

We may replace  pq  with  p + (−q).

The rule of symmetry also means that in any equation, we may exchange the sides.

 If 15 = 2x + 7, then we are allowed to write 2x + 7 = 15.

And so the rules of algebra tell us what we are allowed to write.  They tell us what is legal.

Problem 1.   Use the rule of symmetry to rewrite each of the following. And note that the symmetric version is also a rule of algebra.

Do the problem yourself first!

 a) 1· x  =  x x = 1· x b) (−1)x  =  −x −x = (−1)x c) x + 0  =  x x = x + 0 d) 10  =  3x + 1 3x + 1 = 10
 e) xy = axay axay = xy f) x + (−y) = x − y x − y = x + (−y)
 g) a2 + b2 = a + b   2 a + b   2 = a2 + b2

The commutative rules

The order in which we write terms will not affect the sum.  We express this in algebra by writing

a + b   =  b + a

That is called the commutative rule of addition.  It will apply to any number of terms.

a + b − c + d  =  b + d + a − c  =  −c + a + d + b.

The order does not matter.

Example 1.   Apply the commutative rule to  pq.

Solution.   The commutative rule for addition is stated for the operation + .  Here, though, we have the operation − .  But we can write

 p − q = p + (−q). Therefore, p − q = −q + p.

*

Here is the commutative rule of multiplication:

ab  =  ba

The order of factors does not matter.

abcd  = dbac  = cdba.

The rule applies to any number of factors.

What is more, we may associate factors in any way:

(abc)d = b(dac) = (ca)(db).

And so on.

Example 2.   Multiply  2x · 3y · 5z.

Solution.   The problem means:  Multiply the numbers, and rewrite the letters.

2x · 3y · 5z = 2 · 3 · 5xyz = 30xyz.

It is the style in algebra to write the numerical factor to the left of the literal factors.

Problem 2.   Multiply.

 a) 3x · 5y  = 15xy b) 7p · 6q = 42pq c) 3a · 4b · 5c = 60abc

Problem 3.   Rewrite each expression by applying a commutative rule.

 a) −p + q  = q + (−p) = q − p b) (−1)6 = 6(−1)
 c) (x − 2) + (x + 1) = (x + 1) + (x − 2) d) (x − 2)(x + 1) = (x + 1)(x − 2)

Zero

We have seen the following rule for 0 (Lesson 3 ):

For any number a:

a + 0 = 0 + a = a

0 added to any number  does not change the number.  0 is therefore called the identity of addition.

The inverse of an operation  undoes that operation.

5 + 4,

then to undo that -- to get back to 5 -- we must add −4:

5 + 4 + (−4)  =  5 + 0  =  5.

Adding −4 is the inverse of adding 4, and vice-versa.  We say that −4 is the additive inverse of 4.

In general, corresponding to every number a there is a unique number −a, such that

a + (−a)  =  (−a) + a   =  0

A number combined with its inverse gives the identity.

We have seen that that rule is essentially the definition of −a.

Thus, the additive inverse of a is −a.  And the additive inverse of −a is a.

−(−a) = a.

Problem 4.   Transform each of the following according to a rule of algebra.

 a) xyz + 0 = xyz b) 0 + (−q) = −q c) −¼ + 0 = −¼ d) ½ + (−½) = 0 e) −pqr + pqr = 0 f) x + abc −abc = x

g)  sin x + cos x + (−cos x) = sin x

The student might think that this is trigonometry, but it is not.
g) It is algebra Problem 5 .   Complete the following.

 a) pq + (−pq) = 0 b) z  + (−z) = 0 c) −&2\$ + &2\$ = 0 d) ½x + 0 = ½x e) 0 + (−qr) = −qr f) −π + 0 = −π

g)  tan x + cot x + (−cot x) = tan x.

Two rules for equations

An equation is a statement that two things -- the two sides -- are equal. Inherent in the meaning of equal is the fact that, as long as we do the same thing to both sides, they will still be equal. That is expressed in the following two rules.

 Rule 1. If a = b, then a + c = b + c.

The rule means:

We may add the same number to both sides of an equation.

This is the algebraic version of the axiom of arithmetic and geometry:

If equals are added to equals, the sums are equal.

 Example 3. If x − 2 = 6, then x = 6 + 2 = 8.

-- upon adding 2 to both sides.

 Example 4. If x + 2 = 6, then x = 6 − 2 = 4.

-- upon subtracting 2 from both sides.

But the rule is stated in terms of addition.  Why may we subtract?

Because subtraction is equivalent to addition of the negative.

ab = a + (−b).

Therefore, any rule for addition is also a rule for subtraction.

Note:  In Example 3, adding 2 is the inverse of subtracting 2. And the effect is to transpose −2 to the other side of the equation as +2.

In Example 4, the effect of subtracting 2 from both sides is to transpose +2 to the other side of the equation as −2.

We will go into this more in Lesson 9.

Problem 6.

 a) If b) If x − 1 = 5, x + 1 = 5, then then x = 6. x = 4. On adding 1 to both sides. On subtracting 1 from both sides. c) If d) If x − 4 = −6, x + 4 = −6, then then x = −2. x = −10. On adding 4 to both sides. On subtracting 4 from both sides.
 Rule 2. If a = b, then ca = cb.

This rule means:

We may multiply both sides of an equation by the same number.

Example 5.   If

 2x = 3, then 10x = ?

Now, what happened to 2x to make it 10x ?

We multiplied it by 5.  Therefore, to preserve the equality, we must multiply 3 by 5, also.

10x = 15.

Example 6.   If

 x2 = 5, then x = 10.

Here, we multiplied both sides by 2, and the 2's simply cancel.

See Lesson 26 of Arithmetic, Example 5.

Example 7.   If

 2x = 14, then x = 7.

Here, we divided both sides by 2.  But the rule states that we may multiply both sides.  Why may we divide?

Because division is equal to multiplication by the reciprocal.  In this example, we could say that we multiplied both sides by ½.

Any rule for multiplication, then, is also a rule for division.

Problem 7.

 a) If b) If x = 5, x = −7, then then 2x = 10. −4x = 28. c) If d) If x3 = 2, x4 = −2 then then x = 6. x = −8. On multiplying both sides by 3. On multiplying both sides by 4.
 Problem 8.  Divide both sides. a) If b) If 3x = 12, −2x = 14, then then x = 4. x = −7. On dividing both sides by 3. On dividing both sides by −2. c) If d) If 6x = 5, −3x = −6, then then x = 56 x = 2.

Problem 9.  Changing signs on both sides.  Write the line that results from multiplying both sides by −1.

 a) −x = 5. b) −x = −5. c) −x = 0. x = −5. x = 5. x = −0 = 0.

This problem illustrates the following theorem:

In any equation we may change the signs on both sides.

 If −a = b, then a = −b.

This follows directly from the uniqueness of the additive inverse.

 If −a = b, then a + b = 0. But that implies a = −b.

Which is what we wanted to prove.

We will have occasion to apply this theorem when we come to solve equations.  For we will see that to "solve" an equation we must isolate x -- not −x -- on the left of the equal sign.  And when we come to the distributive rule (Lesson 14), we will see that we may change all the signs on both sides.

Problem 10.

 a) If  x = 9,  then  −x = −9. b) If  x = −9,  then  −x = 9. c) If  −x = 2,  then  x = −2. d) If  −x = −2  then  x = 2.

x is a variable. It is neither positive nor negative. Only numbers are positive or negative. When x takes a value -- positive or negative -- the values of x and −x will have opposite signs. If x takes a positive value, then −x will be negative. But if x takes a negative value, then −x will be positive.

Thus if x = −2, then −x = −(−2) = +2. (Lesson 2.)

(If x = 0, then −x = −0, which we must say is equal to 0. −0 = +0 = 0.) Next Lesson:  Reciprocals and zero

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