Determine whether the system of linear equations has Infinite solution calculator

SolutionHelp

$2x+5y=16,3x+y=11$

$2x+5y=16,4x+10y=32$

$2x+2y+z=5,x-y+z=1,3x+y+2z=4$

$2x+y+z=5,3x+5y+2z=15,2x+y+4z=8$

Example

1. Determine whether the system of linear equations x+y+z=3,2x-y-z=3,x-y+z=9 has No solution

Solution:
Here

$x+y+z=3$

$2x-y-z=3$

$x-y+z=9$

$|D|$

$1$	$1$	$1$
$2$	$-1$	$-1$
$1$	$-1$	$1$

$1$

	$-1$	$-1$
	$-1$	$1$

$-1$

	$2$	$-1$
	$1$	$1$

$+1$

	$2$	$-1$
	$1$	$-1$

$=1 xx (-1 × 1 - (-1) × (-1)) -1 xx (2 × 1 - (-1) × 1) +1 xx (2 × (-1) - (-1) × 1)$

$=1 xx (-1 -1) -1 xx (2 +1) +1 xx (-2 +1)$

$=1 xx (-2) -1 xx (3) +1 xx (-1)$

$= -2 -3 -1$

$=-6$

$|D_1|$

$3$	$1$	$1$
$3$	$-1$	$-1$
$9$	$-1$	$1$

$3$

	$-1$	$-1$
	$-1$	$1$

$-1$

	$3$	$-1$
	$9$	$1$

$+1$

	$3$	$-1$
	$9$	$-1$

$=3 xx (-1 × 1 - (-1) × (-1)) -1 xx (3 × 1 - (-1) × 9) +1 xx (3 × (-1) - (-1) × 9)$

$=3 xx (-1 -1) -1 xx (3 +9) +1 xx (-3 +9)$

$=3 xx (-2) -1 xx (12) +1 xx (6)$

$= -6 -12 +6$

$=-12$

$|D_2|$

$1$	$3$	$1$
$2$	$3$	$-1$
$1$	$9$	$1$

$1$

	$3$	$-1$
	$9$	$1$

$-3$

	$2$	$-1$
	$1$	$1$

$+1$

	$2$	$3$
	$1$	$9$

$=1 xx (3 × 1 - (-1) × 9) -3 xx (2 × 1 - (-1) × 1) +1 xx (2 × 9 - 3 × 1)$

$=1 xx (3 +9) -3 xx (2 +1) +1 xx (18 -3)$

$=1 xx (12) -3 xx (3) +1 xx (15)$

$= 12 -9 +15$

$=18$

$|D_3|$

$1$	$1$	$3$
$2$	$-1$	$3$
$1$	$-1$	$9$

$1$

	$-1$	$3$
	$-1$	$9$

$-1$

	$2$	$3$
	$1$	$9$

$+3$

	$2$	$-1$
	$1$	$-1$

$=1 xx (-1 × 9 - 3 × (-1)) -1 xx (2 × 9 - 3 × 1) +3 xx (2 × (-1) - (-1) × 1)$

$=1 xx (-9 +3) -1 xx (18 -3) +3 xx (-2 +1)$

$=1 xx (-6) -1 xx (15) +3 xx (-1)$

$= -6 -15 -3$

$=-24$

Here

$D=-6,D_1=-12,D_2=18,D_3=-24$

Here,

$D!=0$

Hence, given system has unique solution (System of equation is consistent)