6
SOME RULES
OF
ALGEBRA
ALGEBRA, we can say, is a body of rules. They are rules that show how something written one way may be rewritten a different way. For, what is a calculation if not transforming one set of symbols into another? In arithmetic, we may replace the symbols '2 + 2' with the symbol '4.' In algebra, we may replace 'a + b' with 'b + a.'
Here are some of the basic rules:
| 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
| (−1)a | = | −a, | |
| then -- on exchanging sides -- we may write | |||
| −a | = | (−1)a. | |
This tells us that we may replace the algebraic sign minus (−) with the factor (−1).
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. | |
Thus 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 commutative rules
The order of terms does not matter. We express this in algebra by writing
a + b = b + a
This is called the commutative rule of addition. It will apply to any number of terms. The order does not matter.
a + b − c + d = b + d + a − c = −c + a + d + b
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:
a· b = b· a
This tells us that the order of factors does not matter.
abcd = dbac = cdba
Example 2. Multiply 2x· 3y· 5z.
Solution. The problem means: Multiply the numbers, and rewrite the literal factors.
2x· 3y· 5z = 2· 3· 5xyz = 30xyz
It is the style in algebra to write the numerical factors 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 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. The rule is:
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 thing 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. | ||
This 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, | |||
| then | |||||
| x + 4 | = | 6 | |||
-- upon adding 4 to both sides.
| Example 4. | If | ||||
| x | = | 9, | |||
| then | |||||
| x − 4 | = | 5 | |||
-- upon subtracting 4 from both sides.
But the rule is stated in terms of addition. Why may we subtract?
Because subtracting is equivalent to adding the inverse.
a − b = a + (−b) (Lesson 3)
Subtracting b is the same as "adding" −b.
Therefore, any rule for addition is also a rule for subtraction.
| Problem 6. | ||||||||||
| a) | If | b) | If | |||||||
| x | = | 2, | x | = | 10, | |||||
| then | then | |||||||||
| x + 6 | = | 8. | x − 1 | = | 9. | |||||
| c) | If | d) | If | |||||||
| x | = | −6, | x | = | −2, | |||||
| then | then | |||||||||
| x + 2 | = | −4. | x − 3 | = | −5. | |||||
| 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, if we multiply 3 by 5 also, the equality remains.
10x = 15.
Example 6. If
| 4x | = | 14, | |
| then | |||
| 2x | = | 7. | |
In this example, we have divided both sides by 2. But the Rule states that we may multiply both sides. Why may we divide?
Because division in algebra is equivalent to multiplication by the reciprocal. (Lesson 5.) In this example, we could say that we have multiplied both sides by ½. Therefore, any rule for multiplication is also a rule for division.
Example 7. If
| abx | = | ac, | |
| then | |||
| bx | = | c. | |
On dividing both sides by a, we say that we have "canceled" the a's. In other words,
If both sides of an equation have a common factor,
then we may "cancel" them.
| Problem 7. | ||||||||||
| a) | If | b) | If | |||||||
| x | = | 5, | x | = | −7, | |||||
| then | then | |||||||||
| 2x | = | 10. | −4x | = | 28. | |||||
| c) | If | d) | If | |||||||
| 3x | = | 2, | −5x | = | 1, | |||||
| then | then | |||||||||
| 18x | = | 12. | 25x | = | −5. | |||||
| Problem 8. Divide both sides. | ||||||||||
| a) | If | b) | If | |||||||
| 3x | = | 12, | 10x | = | −15, | |||||
| then | then | |||||||||
| x = 4. | 2x = −3. | |||||||||
| c) | If | d) | If | |||||||
| ½ax | = | ½b, | pqrx | = | 8q, | |||||
| then | then | |||||||||
| ax = b. | prx = 8. | |||||||||
Problem 9. Changing signs. Write the line that results from multiplying each side by −1.
| a) | x | = | 5 | b) | x | = | −5 | c) | −x | = | 5 | d) | −x | = | −5 | |||
| −x | = | −5 | −x | = | 5 | x | = | −5 | x | = | 5 | |||||||
This problem illustrates the following theorem:
In any equation we may change the signs on both sides.
We will see this 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. | |
Next Lesson: Removing grouping symbols
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