II. Simple Equations

Simple Equations.

43. Equations.

An equation is a statement in symbols that two expressions stand for the same number.

Thus, the equation $3 x + 2 = 8$ states that $3 x + 2$ and $8$ stand for the same number.

That part of the equation which precedes the sign of equality is called the first member, or left side, and that which follows the sign of equality is called the second member, or right side.

The statement of equality between two algebraic expressions, if true for all values of the letters involved, is called an identical equation; but if true only for certain particular values of the letters involved, it is called an equation of condition.

Thus, $a + b = b + a$ , which is true for all values of $a$ and $b$ , is an identical equation, and $3 x + 2 = 8$ , which is true only when $x$ stands for $2$ , is an equation of condition.

For brevity, an identical equation is called an identity, and an equation of condition is called simply an equation.

We often employ an equation to discover an unknown number from its relation to known numbers. We usually represent the unknown number by one of the last letters of the alphabet, as $x$ , $y$ , $z$ ; and by way of distinction, we use the first letters, $a$ , $b$ , $c$ , etc., to represent numbers that

are supposed to be known, though not expressed in the number-symbols of Arithmetic.

Thus, in the equation $a x + b = c$ , $x$ is supposed to represent an unknown number, and $a$ , $b$ , and $c$ are supposed to represent known numbers.

47. Simple Equations.

An equation which contains the first power of $x$ , the symbol for the unknown number, and no higher power, is called a simple equation, or an equation of the first degree.

Thus, $a x + b = c$ is a simple equation, or an equation of the first degree in $x$ .

48. Solution of an Equation.

To solve an equation is to find the unknown number; that is, the number which, when substituted for its symbol in the given equation, renders the equation an identity. This number is said to satisfy the equation, and is called the root of the equation.