Using Cramer's rule, solve the following equations
3x=y+z ; 2x+y-z=4 ; x-y+z = 3
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To solve the system by Cramer's rule, we do the following steps:
1. Write the system as a matrix equation AX=b where:
A=
⎣
⎢
⎢
⎡
3
2
4
1
−4
1
1
3
−3
⎦
⎥
⎥
⎤
X=
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
b=
⎣
⎢
⎢
⎡
2
−1
−11
⎦
⎥
⎥
⎤
2. Next, we find the following determinants:
Δ=
∣
∣
∣
∣
∣
∣
∣
∣
3
2
4
1
−4
1
1
3
−3
∣
∣
∣
∣
∣
∣
∣
∣
=∣A∣=63
Δ
1
=
∣
∣
∣
∣
∣
∣
∣
∣
2
−1
−11
1
−4
1
1
3
−3
∣
∣
∣
∣
∣
∣
∣
∣
=−63
Δ
2
=
∣
∣
∣
∣
∣
∣
∣
∣
3
2
4
2
−1
−11
1
3
−3
∣
∣
∣
∣
∣
∣
∣
∣
=126
Δ
3
=
∣
∣
∣
∣
∣
∣
∣
∣
3
2
4
1
−4
1
2
−1
−11
∣
∣
∣
∣
∣
∣
∣
∣
=189
3. The solution is now given by:
x=
Δ
Δ
1
=−1
y=
Δ
Δ
2
=2
z=
Δ
Δ
3
=3
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