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6
SOME RULES
OF
ALGEBRA
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 transforming one set of symbols into another? In arithmetic we may replace the symbols '2 + 2' with this symbol '4.' In algebra, we may replace 'a + (−b)' with 'a − b.'
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 p − q 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.
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| 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) | x y |
= | ax ay |
ax ay |
= | x y |
f) | x + (−y) = x − y | x − y = x + (−y) |
| g) | a 2 |
+ | b 2 |
= | a + b 2 |
a + b 2 |
= | a 2 |
+ | b 2 |
The order of terms does not matter. 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 p − q.
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) |
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 adding
The inverse of an operation undoes that operation.
If we start with 5, for example, and then add 4,
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 −a is the additive inverse of a. Here is the rule:
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.
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. It is
g) 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
| 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 | = | 8 | |||
-- upon adding 2 to both sides.
| Example 4. | If | ||||
| x + 2 | = | 6, | |||
| then | |||||
| x | = | 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.
a − b = a + (−b).
Therefore, any rule for addition is also a rule for subtraction.
| 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
| x 2 |
= | 5, | |
| then | |||
| x | = | 10. | |
Here, we multiplied both sides by 2, and the 2's simply cancel.
See Lesson 26 of Arithmetic, Example 7.
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 equivalent to multiplication by the reciprocal. (Lesson 5.) In this example, we could say that we multiplied both sides by ½.
Therefore, any rule for multiplication 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 | |||||||
| x 3 |
= | 2, | x 4 |
= | −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 | = |
5 6 |
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.
We will see that 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: Removing grouping symbols
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Copyright © 2012 Lawrence Spector
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