5
SOME RULES
OF
ALGEBRA
ALGEBRA, we can say, is a body of rules that show how something in one form may be rewritten in a different form. For, what is a calculation if not replacing 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 basic rules:
| −(−a) | = | a | (Lesson 2) |
| a + (−b) | = | a − b | (Lesson 3) |
| a − (−b) | = | a + b | (Lesson 3) |
| 1· a | = | a | |
| (−1)a | = | −a | |
Associated with these -- and with any rules -- is the rule of symmetry:
If a = b, then b = a.
This means, for one thing, that the rules of algebra go 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. | |
The rules of algebra tell us what we are allowed to write.
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.
Inverses
The inverse of a number -- with respect to 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
With respect to addition, −4 is the inverse of 4. In general, the "additive inverse" of a is −a. The rule is:
a + (−a) = (−a) + a = 0
A number combined with its inverse gives the identity.
Problem 4. Rewrite 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) | abc −bac = 0 | ||
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 | ||
Two rules for equations
| Rule 1. | If | |||
| a | = | b, | ||
| then | ||||
| a + c | = | b + c. | ||
This means,
We may add the same number to both sides of an equation.
| 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 4 is the same as "adding" −4.
Therefore, any rule for addition is also a rule for subtraction.
| Rule 2. | If | |||
| a | = | b, | ||
| then | ||||
| ca | = | cb. | ||
This 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 ?
It was multiplied by 5. Therefore, if we multiply 3 by 5 also, the equality remains.
10x = 15.
| 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. | |||||
| 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. 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 of an equation.
Problem 9.
| 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: Reciprocals and zero
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