First slide

System of linear equations

Question

$The system of linear equations x_{1} + 2 x_{2} + x_{3} = 3, 2 x_{1} + 3 x_{2} + x_{3} = 3 and$ $3 x_{1} + 5 x_{2} + 2 x_{3} = 1 has$

Moderate

A

$I n f i n i t e n u m b e r o f s o l u t i o n s$
B

$e x a c t l y t h r e e s o l u t i o n s$
C

$u n i q u e s o l u t i o n$
D

$N o s o l u t i o n$

Solution

$[A B] = [\begin{array}{l} 1 2 1 3 \\ 2 3 1 3 \\ 3 5 2 1 \end{array}]$
$R_{2} \to R_{2} - 2 R_{1}, R_{3} \to R_{3} - 3 R_{1}$
$~$ $[\begin{array}{l} 1 2 1 3 \\ 0 - 1 - 1 - 3 \\ 0 - 1 - 1 - 8 \end{array}]$
$R_{3} \to R_{3} - R_{2}$
$~ [\begin{array}{l} 1 2 1 3 \\ 0 - 1 - 1 - 3 \\ 0 0 0 - 5 \end{array}]$
$ρ (A) = 2 ρ (A B) = 3$
$ρ (A) \neq ρ (A B)$
no solution

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