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# Solve the following equations using Cramer’s Rule: 2x-1y+3z=4,1x-1y+1z=2,3x+1y-1z = 2 - Mathematics and Statistics

Sum

Solve the following equations using Cramer’s Rule:

2/x - 1/y + 3/z = 4, 1/x - 1/y + 1/z = 2, 3/x + 1/y - 1/z  = 2

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#### Solution

Let 1/x = "p", 1/y = "q", 1/z = r

∴ The given equations become
2p – q + 3r = 4
p – q – r = 2
3p + q – r = 2

D = |(2, -1, 3),(1, -1, 1),(3, 1, -1)|

= 2(1 – 1) – (– 1)(– 1 – 3) + 3(1 + 3)
= 0 – 4 + 12
= 8

Dp = |(4, -1, 3),(2, -1, 1),(2, 1, -1)|

= 4(1 – 1) – (– 1)(– 2 –2) + 3(2 + 2)
= 0 – 4 + 12
= 8

Dq = |(2, 4, 3),(1, 2, 1),(3, 2, -1)|

= 2(–2 – 2) –4(– 1 – 3) + 3(2 – 6)
= – 8 + 16 – 12
= – 4

Dr = |(2, -1, 4),(1, -1, 2),(3, 1, 2)|

= 2(– 2 – 2) – (– 1)(2 – 6) + 4(1 + 3)
= – 8 – 4 + 16
= 4
By Cramer's Rule,

p = "D"_"p"/"D" = 8/8 = 1

q = "D"_"q"/"D" = (-4)/8 = (-1)/2

r = "D"_"r"/"D" = 4/8 = 1/2

∴ 1/x = 1, 1/y = (-1)/2, 1/z = 1/2

∴ x = 1, y = – 2 and z = 2 are the solutions of the given equations.

Concept: Application of Determinants - Cramer’s Rule
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