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nydus/The First Steps in AlgebraPublic

This textbook introduces the fundamental concepts of algebra, including definitions of units, quantities, and number symbols. It explains the use of letters to represent numbers and outlines the signs for basic mathematical operations.

Page 25 of 227
Table of Contents

II.

49. Axioms.

In solving an equation, we make use of the following axioms:

  1. If equal numbers be added to equal numbers, the sums will be equal.
  1. If equal numbers be subtracted from equal numbers, the remainders will be equal.
  1. If equal numbers be multiplied by equal numbers, the products will be equal.
  1. If equal numbers be divided by equal numbers, the quotients will be equal.

If, therefore, the two sides of an equation be increased by, diminished by, multiplied by, or divided by equal numbers, the results will be equal.

Thus, if 8x=24, then 8x+4=24+4, 8x−4=24−4, 4×8x=4×24, and 8x÷4=24÷4.

50. Transposition of Terms.

It becomes necessary in solving an equation to bring all the terms that contain the symbol for the unknown number to one side of the equation, and all the other terms to the other side. This is called transposing the terms. We will illustrate by examples:

  1. Find the number for which x stands when 14x−11=5x+70.

The first object to be attained is to get all the terms which contain x on the left side of the equation, and all the other terms on the right side. This can be done by first subtracting 5x from both sides (Ax. 2), which gives 9x−11=70, and then adding 11 to these equals (Ax. 1), which gives

9x + 11 - 11 = 70 + 11. lint Combine, 9x = 81. lint Divide by 9, x = 9.

  1. Find the number for which x stands when x+b=a.

lint The equation is x + b = a. lint Subtract b from each side, x + b - b = a - b. rint(Ax. 2)

Since +b and −b in the left side cancel each other (§ 30), we have x=a−b.

  1. Find the number for which x stands when x−b=a.

lint The equation is x - b = a. lintAdd +b to each side, x + b - b = a + b. rint(Ax. 1)

Since +b and −b in the left side cancel each other (§ 30), we have x=a+b.

The effect of the operation in the preceding equations, when Axioms (1) and (2) are used, is to take a term from one side and put it on the other side with its sign changed. We can proceed in a like manner in any other case. Hence the general rule:

Any term may be transposed from one side of an equation to the other, provided its sign is changed.

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