108 Cramer’s rule

curves he considers, Cramer presents effective tech-

niques for drawing their graphs. In the second chapter

he discusses the role of geometric transformations as a

means of simplifying the equations to curves (akin to

today’s approach using the notion of

PRINCIPAL AXES

).

This leads to his famous classification of curves in the

third chapter. Here Cramer also discusses the problem

of finding the equation of a degree-two curve ax2+ bxy

+ cy2+ dx + ey = 0 that passes through five previously

specified points in the plane. Substituting in the values

of those points leads to five linear equations in the five

unknowns a, b, c, d, and e. To solve the problem

Cramer then refers the reader to an appendix of the

text, and it is here that his famous rule for solving sys-

tems of equations appears. Cramer made no claim to

the originality of the result and may have been well

aware that Colin Maclaurin had first established the

famous theorem. (Cramer cited the work of Colin

Maclaurin in many footnotes throughout his text, sug-

gesting that he was working closely with the writings

of Maclaurin.)

Cramer also served in local government for many

years, offering expert opinion on matters of artillery

and defense, excavations, and on the reconstruction

and preservation of buildings. He died on January 4,

1752, in Bagnols-sur-Cèze, France. Although Cramer

did not invent the rule that bears his name, he deserves

recognition for developing superior notation for the

rule that clarified its use.

Cramer’s rule Discovered by Scottish mathematician

C

OLIN

M

ACLAURIN

(1698–1746), but first published

by Swiss mathematician G

ABRIEL

C

RAMER

(1704–52),

Cramer’s rule uses the

DETERMINANT

function to find a

solution to a set of

SIMULTANEOUS LINEAR EQUATIONS

.

An example best illustrates the process.

Consider the set of equations:

2x+ 3y+ z= 3

x– 2y+ 2z= 11

3x+ y– 2z= –6

Set Ato be the matrix of coefficients:

The determinant of this matrix is not zero: det(A) = 35.

A standard property of determinants asserts that

if the elements of the first column are multiplied by

the value x, then the determinant changes by the fac-

tor x:

Adding a multiple of another column to the first does

not change the value of the determinant. We shall add y

times the second column, and ztimes to the third to

this first column:

But of course this first column equals the column of

values of the simultaneous equations:

This tells us that the value of xwe seek is:

In the same way, the value yis found as the ratio of the

determinant of the matrix Awith the second column

replaced by the column of values of the simultaneous

equations and the determinant of A: