Cramer’s Rule

Cramer’s rule is a method for solving linear simultaneous equations. It makes use of determinants and so a knowledge of these is necessary before proceeding.

1. Cramer’s Rule – two equations
If we are given a pair of simultaneous equations a1 x + b1 y = d1 a2 x + b2 y = d2 then x, and y can be found from d1 b1 d2 b2 a1 b1 a2 b2 a1 d1 a2 d2 a1 b 1 a2 b 2

x=

y=

Example Solve the equations 3x + 4y = −14 −2x − 3y = 11

Solution Using Cramer’s rule we can write the solution as the ratio of two determinants. −14 4 11 −3 3 4 −2 −3 −2 = 2, −1 3 −14 −2 11 3 4 −2 −3

x=

=

y=

=

5 = −5 −1

The solution of the simultaneous equations is then x = 2, y = −5.

5.2.1

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2. Cramer’s rule – three equations
For the case of three equations in three unknowns: If a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3

then x, y and z can be found from d1 d2 d3 a1 a2 a3 b1 b2 b3 b1 b2 b3 c1 c2 c3 c1 c2 c3 a1 a2 a3 a1 a2 a3 d1 d2 d3 b1 b2 b3 c1 c2 c3 c1 c2 c3 a1 a2 a3 a1 a2 a3 b1 b2 b3 b1 b2 b3 d1 d2 d3 c1 c2 c3

x=

y=

z=

Exercises Use Cramer’s rule to solve the following sets of simultaneous equations. a)

7x + 3y = 15 −2x + 5y = −16

b)

x + 2y + 3z = 17 3x + 2y + z = 11 x − 5y + z = −5

Answers a) x = 3, y = −2.

b) x = 1, y = 2, z = 4