First slide

System of linear equations

Question

The system of linear equations $x - y + z = 1, x + y - z = 3, x - 4 y + 4 z = α$ has

Moderate

A

A unique solution when $α = 2$
B

A unique solution when $α \neq 2$
C

An infinitely number of solutions when $α = 2$
D

An infinitely number of solutions when $α = - 2$

Solution

given $x - y + z = 1(1)$
$x + y - z = 3(2)$
$x - 4 y + 4 z = α__(3)$
From eqns 1 & 2 $x = 2 \Rightarrow$ y-z=1 & $- 4 y + 4 z = α - 2$
$4 y - 4 z + (- 4 y + 4 z) = 4 + (α - 2) \Rightarrow α + 2 = 0 \Rightarrow α = - 2$
The system of has no solution for $α \neq - 2$
If $α = - 2,$ then the system is consistent and has infinite number of solutions. (since y-z=1 has infinite number of solutions)

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