S
IMPSON
’
S RULE
, as it is called today, appeared in
his 1743 text Mathematical Dissertations. That his
name became attached to a technique that he clearly did
not invent, but merely described, serves as a testimony
to the influence and popularity of his writing. Simpson
died in Market Bosworth, England, on May 14, 1761.
Simpson’s rule See
NUMERICAL INTEGRATION
.
simultaneous linear equations An equation is
called linear if each term of the equation containing a
variable contains just one variable raised to the first
power. For example, 3x+ 4 = 6 and 2x+ 7y– z= 11
are linear equations (but 2xy + z= 5 and 3x2+ 4 = 6,
for instance, are not). A collection of linear equations
in several variables required to simultaneously hold
true for some value of those unknowns is called a sys-
tem of simultaneous linear equations. For example,
the equations
5x+ 2y= 16
2x– y= 1
form a pair of simultaneous linear equations. This pair
has the solution x= 2 and y= 3.
There are a number of methods for solving pairs of
simultaneous linear equations.
Elimination Method: Multiplying each equation by a
suitable quantity so that the coefficients of one of
the variables match. The solution to the system is
then readily apparent if one subtracts the two equa-
tions. For example, to solve
x+ 3y= 7
3x– y= 11
multiply the first equation through by 3 to obtain
the equivalent pair:
3x+ 9y= 21
3x– y= 11
Subtracting yields the equation: (3x+ 9y) – (3x– y)
= 21 – 11, that is, 10y= 10, indicating that yequals
1. Since x+ 3y= 7 we now see that xmust be 4.
Substitution Method: Solve for one of the variables in
one equation and insert the result into the second
equation. For example, for the pair
x+ 3y= 7
3x– y= 11
the first equation gives x= 7 – 3y. Substituting into the
second equation yields 3(7 – 3y) – y= 11, that is, 21 –
10y= 11, or, again, that 10y= 10. The solution to the
system follows as shown in the previous method.
Method of Equating: Solve both equations for one
variable and equate the results. In our example, the
first equation yields x= 7 – 3yand the second,
x= . Equating yields 7 – 3y= or
21–9y= 11 + y. Solving for ygives y= 1, from
which it follows that x= 4.
One can also seek a
GRAPHICAL SOLUTION
. Such an
approach shows that any pair of simultaneous linear
equations either has a unique solution (as for the pair
above), no solution, or infinitely many solutions. It is
impossible for a pair of linear equations to have exactly
two distinct solutions, for instance.
The process of G
AUSSIAN ELIMINATION
provides
the means to solve systems of linear equations with
more than two unknowns. This approach yields key
results about the theory of matrices and the study of
LINEAR ALGEBRA
.
See also
HISTORY OF EQUATIONS AND ALGEBRA
(essay);
MATRIX
;
SYSTEM OF EQUATIONS
.
singular point (singularity) Any point on a curve at
which there is not a single well-defined
TANGENT
to the
curve is called a singular point. It may be that the curve
crosses itself at that point, in which case there are two
tangents to the curve, or that the point is a
CUSP
or an
ISOLATED POINT
, for example.
See also
DOUBLE POINT
.
SI units Adopted in 1960 by international agreement,
the Système International d’Unités, (SI units) is a coher-
ent system of units used for scientific purposes. It is
based on seven fundamental base units of which the
meter (m), the kilogram (kg), and the second (s) for
measuring length, mass, and time, respectively, are the
11 + y
———
3
11 + y
———
3
464 Simpson’s rule