matics. All sorts of mathematical problems consist in the indirect determination of some quantity by means of its relations to other quantities which are known, and these relations are all expressed by means of equations. The operation in general of solving a problem in Mathematics, other than a transformation, is first, to express the conditions of the problem by means of one or more equations, and secondly, to solve these equations. For example, the problem which is expressed by the equation above given is the very simple question, "What is the number such that if multiplied by 3, the product is 6 ? " In the present chapter, it is the second of these two operations, the solution of an equation, that is considered.
76. An equation which involves the unknown quantity in the first degree is called a simple equation.
The process of solving a simple equation depends upon the following axioms :
1. If to equals we add equals, the sums are equal.
2. If from equals we take equals, the remainders are equal.
3. If equals are multiplied by equals, the products are equal.
4. If equals are divided by equals, the quotients are equal.
77. Consider the equation 7x = 14.
It is required to find what numerical value x must have consistent with this statement. Dividing both sides by 7, we get
x= 2 (Axiom 4).
Similarly, if {x }{2}= — 6, multiplying both sides by 2, we get
b= -12 . . . . (Axiom 3).
Again, in the equation 7x— 2 x— x= 23 + 15 — 10, by collecting terms, we have 4 x= 28. x = 7.
