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
x
2 – 3xy + z
4 = 11
3x(y + z) = z
2 (x + y)
x
3 + y
3 + z
3 = 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 x and y for
which x
2 + y
2 = 1 and y
2 + x
2 = 2.)
Solving an arbitrary system of equations, if at all
possible, is usually a very difficult task. Often a mathematician will resort to graphical methods and search
for a GRAPHICAL SOLUTION. In special cases, a method
of SUBSTITUTION might prove useful. For instance, consider the equations:
x
2 + y
2 = 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, x
2 + (x – 1)
2 = 25.
This can be rewritten 2x
2 – 2x + 1 = 25 or as 2x
2 – 2x
– 24 = 0. Solving for x yields two solutions, with corresponding 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 x
2 + y
2 = 25.)
The study of LINEAR ALGEBRA and the process of
GAUSSIAN ELIMINATION provide effective, straightforward means to solve systems of SIMULTANEOUS LINEAR
EQUATIONS.
See also CRAMER’S RULE; HISTORY OF EQUATIONS
AND ALGEBRA (essay).
system of equations 491