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✌5.2

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

1. Cramer’s Rule - two equationsIf we are given a pair of simultaneous equations

a1x + b1y = d1

a2x + b2y = d2

then x, and y can be found from

x =

∣∣∣∣∣

d1 b1

d2 b2

∣∣∣∣∣

∣∣∣∣∣

a1 b1

a2 b2

∣∣∣∣∣

y =

∣∣∣∣∣

a1 d1

a2 d2

∣∣∣∣∣

∣∣∣∣∣

a1 b1

a2 b2

∣∣∣∣∣

ExampleSolve the equations

3x + 4y = −14

−2x− 3y = 11

SolutionUsing Cramer’s rule we can write the solution as the ratio of two determinants.

x =

∣∣∣∣∣

−14 411 −3

∣∣∣∣∣

∣∣∣∣∣

3 4−2 −3

∣∣∣∣∣

=−2

−1= 2, y =

∣∣∣∣∣

3 −14−2 11

∣∣∣∣∣

∣∣∣∣∣

3 4−2 −3

∣∣∣∣∣

=5

−1= −5

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

5.2.1 copyright c© Pearson Education Limited, 2000

2. Cramer’s rule - three equationsFor the case of three equations in three unknowns: If

a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3

then x, y and z can be found from

x =

∣∣∣∣∣∣∣

d1 b1 c1

d2 b2 c2

d3 b3 c3

∣∣∣∣∣∣∣

∣∣∣∣∣∣∣

a1 b1 c1

a2 b2 c2

a3 b3 c3

∣∣∣∣∣∣∣

y =

∣∣∣∣∣∣∣

a1 d1 c1

a2 d2 c2

a3 d3 c3

∣∣∣∣∣∣∣

∣∣∣∣∣∣∣

a1 b1 c1

a2 b2 c2

a3 b3 c3

∣∣∣∣∣∣∣

z =

∣∣∣∣∣∣∣

a1 b1 d1

a2 b2 d2

a3 b3 d3

∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣

a1 b1 c1

a2 b2 c2

a3 b3 c3

∣∣∣∣∣∣∣∣∣

ExercisesUse 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

Answersa) x = 3, y = −2. b) x = 1, y = 2, z = 4

5.2.2 copyright c© Pearson Education Limited, 2000