synthetic division See

NESTED MULTIPLICATION

.

system of equations A set of equations in several

unknowns—required to be true for particular values of

those unknowns—is called a system of equations. For

instance, the three equations

x2– 3xy + z4= 11

3x(y+ z) = z2(x+ y)

x3+ y3+ z3= 17

has the solution x= 1, y= 2, and z= 2. Geometrically,

each equation represents a curve or surface in space,

and a solution to the system of equations is a common

point of intersection of those surfaces. It is possible for

a system of equations to have no solutions. (For

instance, there are clearly no values of xand yfor

which x2+ y2= 1 and y2+ x2= 2.)

Solving an arbitrary system of equations, if at all

possible, is usually a very difficult task. Often a mathe-

matician will resort to graphical methods and search

for a

GRAPHICAL SOLUTION

. In special cases, a method

of

SUBSTITUTION

might prove useful. For instance, con-

sider the equations:

x2+ y2= 25

x– y= 1

Solving for y in the second equation yields y= x–1.

Inserting this value into the first equation gives a

QUADRATIC

equation in x, namely, x2+ (x– 1)2= 25.

This can be rewritten 2x2– 2x+ 1 = 25 or as 2x2– 2x

– 24 = 0. Solving for xyields two solutions, with cor-

responding values for y. We have x= 4 and y= 3 as

one solution, and x= –3 and y= –4 as another.

(Graphically, we have found the location of the two

points at which the straight line x– y= 1 intersects the

CIRCLE

x2+ y2= 25.)

The study of

LINEAR ALGEBRA

and the process of

G

AUSSIAN ELIMINATION

provide effective, straightfor-

ward means to solve systems of

SIMULTANEOUS LINEAR

EQUATIONS

.

See also C

RAMER

’

S RULE

;

HISTORY OF EQUATIONS

AND ALGEBRA

(essay).

system of equations 491