ALGEBRA
Sec. 2
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Márquez
CANCELLATION LAWS
EQUALITY AXIOMS: CANCELLATION LAW OF ADDITION
In solving many math problems we will often be faced with equations that look somewhat like this
x+3=5
It is often the case, that we would like to solve for x, meaning we would like to isolate it all by itself on
one side. Specifically, in this case we would like to abolish, kill and cancel the 3 on the left side. For this
we have the perfect medicine. The additive 3 killer is by definition −3. If only we would add a 3 killer
on both sides we would be able to solve for x.
(x + 3) + −3 = 5 + −3
It is the cancellation law of addition axiom that says we are allowed to
add any real number on both sides and the equality will still hold. It
may be helpful to think of the equation as a balanced balance, so long
as we add the same real number to both sides it will remain in balance.
Stated formally we say:
IF
A=B
THEN
(A) + c = (B) + c
< CLA >
Note, we use the parenthesis to emphasize we are adding to entire side.
This may not have much consequences now, but it will have important
consequences in the case of multiplication [below]. Moreover, we will
assume we can add any number on either side. In the formal statement
above we added ’c’ on both sides on the right, we could also add it like
this:
c + (A) = c + (B)
< CLA >
We will adopt the descriptive symbol
to indicate adding a number to each side of the equation.
EQUALITY AXIOMS: CANCELLATION LAW OF MULTIPLICATION
Now consider the world of multiplication. Suppose we encountered an
equation such as
3x = 5
To solve such equation, we would like to kill, abolish and cancel the 3
which is begin multiplied by the x. To do this we need the multiplicative
killer of 3, also known as the multiplicative inverse of 3, also know as 13 .
1
1
(3x) = (5)
3
3
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2007
MathHands.com
ALGEBRA
Sec. 2
MathHands.com
Márquez
It is the cancellation law of multiplication axiom that says we are allowed to multiply both sides of an
equation by any real number and the equation will still hold. Formally we state as:
IF
THEN
A=B
(A)c = (B)c
< CLM >
ALSO, on can be applied on the left side...
c(A) = c(B)
< CLM >
We should note that when there is more than one term on either side, the cancellation laws state we may
add or multiply the same quantity onto each side, the important implication is that it should be done
to the entire side, not a part of the side. Consider the following sequence of steps, where we attempt to
apply the Cancellation Law of Multiplication, by multiplying both sides by k:
A+B =C
k · A + B = kC
(given)
(CLM)
Here lies a BIG PROBLEM, the right side was multiplied by k, but only a part of the left side was
multiplied by k, namely, the A, and not the B. Thus is is not a correct application of CLM. The correct
application looks like this:
A+B =C
k · (A + B) = kC
(given)
(CLM)
This sort of problem, we will avoid by, so long as we are learning these axioms, always using parenthesis,
when we apply CLA, or CLM, as done above.
We will adopt the descriptive symbol
to indicate adding a number to each side of the equation.
THEOREM: ADD TWO EQUATIONS theorem [ATE]
Our journey will take us to places where we will undoubtably benefit from adding two equations together.
That is adding both sides of the equations as well as the right sides. Using the axioms presenter here,
we can prove this results in an equality. We can prove that the sum of the left sides will be equal to the
sum of the right sides. This theorem will be particularly useful in the near future when solving systems
of equations.
IF A = B
Then A + C = B + D
< AT E >
and C = D
THEOREM: MULTIPLY the TWO EQUATIONS theorem [MTE]
Our journey will take us to places where we will undoubtably benefit from adding two equations together.
That is adding both sides of the equations as well as the right sides. Using the axioms presenter here,
c
2007
MathHands.com
ALGEBRA
Sec. 2
MathHands.com
Márquez
we can prove this results in an equality. We can prove that the sum of the left sides will be equal to the
sum of the right sides. This theorem will be particularly useful in the near future when solving systems
of equations.
IF A = B
Then AC = BD
< MT E >
and C = D
THEOREM: SQUARE BOTH SIDES theorem [SBST]
This theorem will also prove useful in the near future. At this point we have enough axioms and developed
ideas to understand it completely. We present it here and leave as an important excersize to prove it.
IF
A =B
Then
A2 = B 2
< SBST >
c
2007
MathHands.com
ALGEBRA
Sec. 2
MathHands.com
Márquez
1. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
x = 7
(x)+3 = (7)+3
2. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
x = 7
3+(x) = 3+(7)
3. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
A+3·0 = 7
(A + 3 · 0) + −3 · 0 = (7) + −3 · 0
THEN
4. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
A+3·0 = 7
(A + 3 · 0) + −cat = (7) + −cat
5. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
A+3·0 = 7
(A + 3 · 0)(5) = (7)(5)
6. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
3x + 2 = 7
1
1
(3x + 2) =
· (7)
3
3
c
2007
MathHands.com
ALGEBRA
Sec. 2
MathHands.com
Márquez
7. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
3x + 2 = 7
1
1
· 3x + 2 =
·7
3
3
8. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
3x + 2 = 7
1
1
+ (3x + 2) =
+ (7)
3
3
9. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
3x + 2 = 7
1
1
+ 3x + 2 =
+7
3
3
10. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
3x + 2 = 7
−3x + 3x + 2 = −3x + 7
THEN
11. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
3x + 2 = 7
1
1
· 3x + 2 =
·7
3
3
12. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
3x + 2 = 7
1
1
· (3x + 2) =
· (7)
3
3
c
2007
MathHands.com
ALGEBRA
Sec. 2
MathHands.com
Márquez
13. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
3x · 2 = 7
1
1
· (3x · 2) =
· (7)
3
3
14. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
3x · 2 = 7
1
1
· 3x · 2 =
·7
3
3
15. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
3x + 2 = 7
THEN
−3x(3x + 2) = −3x(7)
16. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
3x + 2 = 7
THEN
0(3x + 2) = 0(7)
17. (True, False, WDKY (we don’t know yet)) Do you know why? explain.
IF
THEN
2 = 7
−2 + (2) = −2 + (7)
18. Prove [ATE]
19. Prove [SBST]
20. Prove [MTE]
21. Prove [CbST] ... IF a = b then a3 = b3
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2007
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