Equations & Transposition

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Equations

(1) Absolute equality between two concrete things is not possible.

Two apples, however much alike, will always have differences in some aspects, such as, the internal arrangement of atoms, or the space they occupy.

(2) Equality becomes possible only in terms of abstraction.

One apple may not be equal to another apple, but, when abstracted, the number “one” for an apple may be said to be equal to “one” for another apple.

(3) An equation has two sides separated by an equal sign (=).

A + B   =   C          (where A, B, and C may stand for numbers)

L.H.S. (Left hand side) is A + B

R.H.S. (Right hand side) is C

(4) An equation implies equality of the two sides, such as,

3 + 4  =  7

(5) When the same quantity is added to the two sides of an equation, the equality of the two side remains.

If        A  =  B            then            A + N  =  B + N

(6) When the same quantity is subtracted from the two sides of an equation, the equality of the two side remains.

If        A  =  B            then            A – N  =  B – N

(7) When the two sides of an equation are multiplied by the same quantity, the equality of the two side remains.

If        A  =  B            then            A . N = B . N

Dot (.) Is used for multiplication

(8) When the two sides of an equation are divided by the same quantity, the equality of the two side remains.

If        A  =  B            then            A / N = B / N

Slash (/) Is used for division

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Rules of Transposition

NOTE: The word TRANSPOSITION means “to position across.”

(9) A term added to one side, when moved to the other side, becomes subtracted. Or, a positive quantity, when moved to the other side, becomes a negative quantity.

If        A + C   =   B            then            A   =   B – C 

C is moved from LHS to RHS

If        A   =   B + C            then            A – C   =   B 

C is moved from RHS to LHS

(10) A term subtracted from one side, when moved to the other side, becomes added. Or, a negative quantity, when moved to the other side, becomes a positive quantity.

If        A  C   =   B            then            A   =   B + C 

C is moved from LHS to RHS

If        A   =   B  C            then            A + C   =   B 

C is moved from RHS to LHS

(11) A factor multiplied at one side, when moved to the other side, becomes a divisor.

If        A . C   =   B            then            A   =   B / C 

C is moved from LHS to RHS

If        A   =   B . C            then            A / C   =   B 

C is moved from RHS to LHS

(12) A divisor at one side, when moved to the other side, becomes factor multiplied.

If        A / C   =   B            then            A   =   B . C 

C is moved from LHS to RHS

If        A   =   B / C            then            A . C   =   B 

C is moved from RHS to LHS

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Application of Transposition

(13) If A – B = C then what is B in terms of C and A?

A – B  =  C    then    A   =  C + B    then    A – C   =  B,    or    B   =  A – C

(14) If A / B = C then what is B in terms of C and A?

A / B   =  C    then    A   =  C . B    then    A / C   =  B,    or    B   =  A / C

(15) When working with word problems, we translate the word problem into an equation. We bring the terms with unknown quantities to the LHS (left hand side), and the rest to RHS (right hand side), and then solve for the unknown quantity.

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EXAMPLE: 3 more than 5 times a number is 13. What is that number?
Suppose the number is N. The problem translates to the equation,
5N + 3 = 13
We first transpose the term 3 to the right.
5N = 13 – 3 = 10
Then we transpose the factor 5 to the right.
N = 10 / 5 = 2
Then we verify the equation for the answer.
5 (2) + 3 = 13
The answer N = 2 is correct.

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EXAMPLE: If 5A – 6 = 14, what is the value of A ?
We get,               5A = 14 + 6 = 20               (Transpose 6)
Therefore,           A = 20 / 5 = 4                    (Transpose 5)
Verify by substituting 4 for A in the original relation.
We get,               5 (4) – 6 = 14                    (Verified correct)

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EXAMPLE: Determine the value of A if 7A / 3 + 4 = 18.
We get,              7A / 3 = 18 – 4 = 14          (Transpose 4)
Therefore,          7A = 14 x 3 = 42               (Transpose 3)
Therefore,          A = 42 / 7 = 6                    (Transpose 7)
Verify by substituting 6 for A in the original relation.
We get,               (7)(6) / 3 + 4 = 18            (Verified correct)

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