XI. SIMPLE EQUATIONS.

CONTAINING ONE UNKNOWN QUANTITY.

171. An Equation is a statement that two algebraic expressions (Art. 16) are equal.

The First Member of an equation is the expression to the left of the sign of equality, and the Second Member is the expression to the right of that sign.

Thus, in the equation 2x-3=3x+5, the first member is 2x-3, and the second member is 3x + 5:

Any term of either member of an equation is called a term of the equation.

The sides of an equation are its two members.

172. An Identical Equation, or, Identity, is one whose members are equal, whatever values are given to the letters, if any, involved; as,

(a + b) (a − b) = a2 — b2, or 5 = 5.

Note. The sign =, read “identically equal to," is frequently used in place of the sign of equality in an identity.

173. An equation is said to be satisfied by a set of values of certain letters involved in it when, on substituting the value of each letter in place of the letter wherever it occurs, the equation becomes identical.

Thus, the equation x2-xy = 24 is satisfied by the set of values = 3, y=-5; for on substituting 3 in place of x, and 5 in place of y, the equation becomes

which is identical.

24 24,

174. An Equation of Condition is an equation involving one or more letters, called unknown quantities, which is not satisfied by every set of values of these letters.

Thus, the equation x + 2 = 5 is satisfied only by the value x=3; again, the equation 2xy=-3 is not satisfied by every set of values of x and y, but is satisfied by the set of values = 2, y = 7.

An equation of condition is usually called an equation.

Any letter in an equation of condition may represent an unknown quantity; but it is usual to represent an unknown quantity by one of the final letters of the alphabet, x, y, z.

175. If an equation contains but one unknown quantity, any value of the unknown quantity which satisfies the equation is called a Root of the equation.

The solution of an equation containing but one unknown quantity is the process of finding its roots.

A root is said to be verified when, on substituting the value of the root in place of its symbol, the equation becomes identical.

176. A Numerical Equation is one in which all the known numbers are represented by Arabic numerals; as,

2x-17x – 5.

A Literal Equation is one in which some or all of the known numbers are represented by letters; as,

2x + a = bx2 - 10.

177. A monomial is said to be rational and integral with respect to one or more letters involved in it when it either is, or can be reduced to, a rational and integral expression (Art. 23), or a rational fraction containing the letters only in its numerator when in its lowest terms (Art. 159).

2 a3b2 Thus,

is rational and integral with respect to a.

A polynomial is said to be rational and integral with respect to one or more letters involved in it when each term involving the letters is rational and integral with respect to them.

178. If a monomial is rational and integral with respect to one or more letters involved in it, its degree with respect to them is denoted by the sum of their exponents.

Thus,

2 a3b'd

is of the third degree with respect to a, and

of the fifth degree with respect to a and b.

The degree with respect to one or more letters of a polynomial which is rational and integral with respect to them, is the degree with respect to these letters of the term or terms of highest degree with respect to them.

to

Thus, 3xy+bx-cy is of the second degree with respect X, and of the third degree with respect to x and y.

179. If each member of an equation is rational and integral with respect to the unknown quantities involved in it, the degree of the equation is the degree with respect to the unknown quantities of the term or terms of highest degree with respect to them.

Thus, if a, b, and c denote known, and x, y, and z unknown quantities,

x - y2z

=

5

degree;

Ɑ= bx + c is an equation of the second degree;

and so on.

=

ax2-b is an equation of the third degree;

A Simple Equation is an equation of the first degree.

PROPERTIES OF EQUATIONS.

180. It follows from Arts. 40, 43, 54, and 69 that:

I. The same quantity may be added to, or subtracted from, both members of an equation without destroying the equality. II. Both members of an equation may be multiplied, or divided, by the same quantity without destroying the equality.