First slide

System of linear equations

Question

$For the equations x + 2 y + 3 z = 1, 2 x + y + 3 z = 2, 5 x + 5 y + 9 z = 4$

Moderate

A

There is only solution
2)
3)
4)
B

There exists infinitely many solutions
C

There is no solution
D

none

Solution

Augmented matrix is
$[\begin{matrix} 1 & 2 & 3 & 1 \\ 2 & 1 & 3 & 2 \\ 5 & 5 & 9 & 4 \end{matrix}]$
$\begin{matrix} R_{2} - 2 R_{1} \\ R_{3} - 5 R_{1} \\ ~ \end{matrix} [\begin{matrix} 1 & 2 & 3 & 1 \\ 0 & - 3 & - 3 & 0 \\ 0 & - 5 & - 6 & - 1 \end{matrix}]$
$\begin{matrix} R_{3} - \frac{5}{3} R_{2} \\ ~ \end{matrix} [\begin{matrix} 3 & - 1 & 1 & 5 \\ 0 & - 2 & 0 & 0 \\ 0 & 0 & - 1 & 0 \end{matrix}]$
there is only one solution

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