To solve for y, we multiply each side of equation (1) by and each side of equation
(2) by and add.
Multiply equation by
Multiply equation by
Add.We can write both numerators and the common denominator of these values for x
and yas determinants because
and
Using these results, the solutions for xand ybecome
where
For convenience, we denote the three determinants in the solution as
Notice that the elements of Dare the four coefficients of the variables in the given
system. The elements of are obtained by replacing the coefficients of xby the
respective constants. Similarly, the elements of are obtained by replacing the coeffi-
cients of yby the respective constants. These results are summarized as Cramer’s
rule.
Dy
Dx
`
a 1
a 2
c 1
c 2
` ` = Dy.
c 1
c 2
b 1
b 2
` ` =Dx, and
a 1
a 2
b 1
b 2
` = D,
`
a 1
a 2
b 1
b 2
y= ` Z0.
`
a 1
a 2
c 1
c 2
`
`
a 1
a 2
b 1
b 2
`
x= ,
`
c 1
c 2
b 1
b 2
`
`
a 1
a 2
b 1
b 2
`
and
a 1 b 2 - a 2 b 1 = `
a 1
a 2
b 1
b 2
c 1 b 2 - c 2 b 1 = ` `.
c 1
c 2
b 1
b 2
a 1 c 2 - a 2 c 1 =^2 `,
a 1
a 2
c 1
c 2
(^2) ,
y = 1 if a 1 b 2 - a 2 b 1 Z 02
a 1 c 2 - a 2 c 1
a 1 b 2 - a 2 b 1
1 a 1 b 2 - a 2 b 12 y=a 1 c 2 - a 2 c 1
a 1 a 2 x+a 1 b 2 y=a 1 c 2 122 a 1.
- a 1 a 2 x-a 2 b 1 y=-a 2 c 1 112 - a 2.
a 1
- a 2
718 APPENDIX A Determinants and Cramer’s Rule
Cramer’s Rule for 2 2 Systems