Determine whether the system of linear equations has Infinite solution calculator

SolutionHelp

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

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

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

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

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 $x+y+z=3$

2x-y-z=3 $2x-y-z=3$

x-y+z=9 $x-y+z=9$

|D| $|D|$

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

1 $1$

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

-1 $-1$

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

+1 $+1$

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

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

=-2-3-1 $= -2 -3 -1$

=-6 $=-6$

|D1| $|D_1|$

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

3 $3$

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

-1 $-1$

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

+1 $+1$

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

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

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

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

=-6-12+6 $= -6 -12 +6$

=-12 $=-12$

|D2| $|D_2|$

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

1 $1$

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

-3 $-3$

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

+1 $+1$

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

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

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

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

=12-9+15 $= 12 -9 +15$

=18 $=18$

|D3| $|D_3|$

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

1 $1$

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

-1 $-1$

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

+3 $+3$

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

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

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

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

=-6-15-3 $= -6 -15 -3$

=-24 $=-24$

Here

D=-6,D1=-12,D2=18,D3=-24 $D=-6,D_1=-12,D_2=18,D_3=-24$

Here,

D≠0 $D!=0$

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