using Cramer's rule solve the given equation. x+y+z=3 2x+3y+4z=9
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Maths
Solve the following equations by Cramer's rule:
3x+
y+
z=
2,2x−
4y+
3z=
−1,4x+
y−
3z=
−11
Answer
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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