equation of a line 163

TRIPLES

. Also, using his famous formula eiθ= cos(θ) +

isin(θ), today called E

ULER

’

S FORMULA

, many trigono-

metric identities can be established quickly by equat-

ing real and imaginary parts. As another application,

consider the series . Since cos(nθ) is the real

part of einθ, the series in question is the real part of

the

GEOMETRIC SERIES

. Evaluating, gives

Thus we have .

equation A mathematical statement that asserts that

one expression or quantity is equal to another is called

an equation. The two expressions or quantities involved

are connected by an equals sign, “=.” For instance, the

statement (a + b)2= a2+ 2ab + b2is an equation, as are

the statements 2x+ 3 = 11 and 10 = 2 ×5.

An equation that is true for all possible values of

the variables involved is called an

IDENTITY

. For

instance, y2– 1 = (y– 1)(y+ 1) is an identity: this equa-

tion is true no matter which value is chosen for y. An

equation that is true only for certain values of the vari-

ables is called a conditional equation. For instance, the

equation 2x+ 3 = 11 is conditional, since it is true only

if xis four. The equation 2c+ d= 6 is also a condi-

tional, since it holds only for certain values of cand d.

The numbers that make a conditional equation

true are called the solutions or roots of the equation.

For example, the solution to 2x+ 3 = 11 is x= 4. An

equation may possess more than one solution, and the

set of all possible solutions to a conditional equation is

called its solution set. For example, the equation y2= 9

has solution set {3, – 3}, and the solution set of the

equation 2c+ d= 6 is the set of all pairs of numbers of

the form (c, 6 – 2c).

The basic principle in solving an equation is to

add, subtract, multiply, or divide both sides of the

equation by the same number until the desired variable

is isolated on one side of the equation. For example,

the equation 2x+ 7 = 5x+ 1 can be solved by subtract-

ing 1 from both sides to give 2x+ 6 = 5x, then sub-

tracting 2xfrom both sides to obtain 6 = 3x, and,

finally, dividing both sides by 3 to obtain the solution x

= 2. This approach works well for

LINEAR EQUATION

s

of one variable. For

QUADRATIC

equations, and

POLY

-

NOMIAL

equations of high degree, one may also be

required to take square and higher roots in the process

of solving the equation. Not all equations, however,

can be solved algebraically, in which case one can seek

only a

GRAPHICAL SOLUTION

.

It should be noted that performing the same arbi-

trary operation on both sides of an equation need not

necessarily preserve the validity of the equation. For

example, although the statement 9 = 9 is certainly

valid, taking a square root on both sides of this trivial

equation could be said to yield the invalid result –3 = 3.

Although 2(x– 1) = 3(x– 1) is true for the value x= 1,

dividing through by the quantity x– 1 yields the

invalid conclusion 2 = 3. And finally, since = ,

selecting the numerator of each side of the equation

yields the absurdity 12 = 3. Care must be taken to

ensure that the operations being used in solving an

equation do indeed preserve equality.

Even if the application of the same operation on

both sides on an equation is deemed valid, such an act

may nonetheless yield a new equation not necessarily

exactly equivalent to the first. For instance, starting with

a= b, squaring both sides yields the equation a2=b2,

which now means a= bor a= –b. Mathematicians use

the symbol “⇒” to denote that one equation leads to a

second, but that the second need not imply the first. For

example, it is appropriate to write: a= b⇒a2= b2. (But

a2= b2⇒

/a= b.) The symbol “⇔” is used to indicate

that two equations are equivalent, that is, that the first

implies the second, and that the second implies the first.

For example, we have: 4x= 12 ⇔x= 3.

See also

CUBIC EQUATION

;

HISTORY OF EQUATIONS

AND ALGEBRA

(essay).

equation of a line A straight

LINE

in two-dimen-

sional space has the property that the ratio of the dif-

ference in y-coordinates of any two points on the line

(rise) to the difference of their x-coordinates (run) is

always the same. That is, the

SLOPE

of a straight line is

constant and can be computed from any two given

points on the line. Precisely, if (a1,b1) and (a2,b2) are

3

–

4

12

–

16