# Solve the following linear equations by using Cramer’s Rule: x + y + z = 6, x – y + z = 2, x + 2y – z = 2 - Mathematics and Statistics

Sum

Solve the following linear equations by using Cramer’s Rule:

x + y + z  = 6, x – y + z = 2, x + 2y – z = 2

#### Solution

Given equations are
x + y + z  = 6,

x – y + z = 2,

x + 2y – z = 2

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

= 1(1 – 2) – 1(–1 – 1) + 1(2 + 1)

= 1 (–1) –1 (–2) + 1(3)

= –1 + 2 + 3

= 4 ≠ 0

Dx = |(6, 1, 1),(2, -1, 1),(2, 2, -1)|

= 6(1 – 2) – 1(–2 – 2) + 1(4 + 2)

= 6(– 4) – 1 (– 4) + 1(6)

= –6 + 4 + 6

= 4

Dy = |(1, 6, 1),(1, 2, 1),(1, 2, -1)|

= 1(–2 – 2) – 6(–1 – 1) + 1(2 – 2)

= 1(–4) – 6(–2) + 1(0)

= –4 + 12 + 0

= 8

Dz = |(1, 1, 6),(1, -1, 2),(1, 2, 2)|

= 1(–2 – 4) – 1(2 – 2) + 6(2 + 1)

= 1(– 6) – 1(0) + 6(3)

= –6 + 0 + 18

= 12

By Cramer’s Rule,

x = "D"_x/"D" = 4/4 = 1,  y = "D"_y/"D" = 8/4 = 2 and

z = "D"_z/"D" = 12/14 = 3

∴ x = 1, y = 2 and z = 3 are the solutions of the given equations.

Concept: Application of Determinants - Cramer’s Rule
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 4 Determinants and Matrices
Exercise 4.3 | Q 1. (i) | Page 74