Determine whether the system of linear equations is inconsistent Examples

1. Determine whether the system of linear equations 2x+y+z=5,3x+5y+2z=15,2x+y+4z=8 is inconsistent

Solution:
Here

2x+y+z=5 $2x+y+z=5$

3x+5y+2z=15 $3x+5y+2z=15$

2x+y+4z=8 $2x+y+4z=8$

|D| $|D|$

2 $2$	1 $1$	1 $1$
3 $3$	5 $5$	2 $2$
2 $2$	1 $1$	4 $4$

2 $2$

	5 $5$	2 $2$
	1 $1$	4 $4$

-1 $-1$

	3 $3$	2 $2$
	2 $2$	4 $4$

+1 $+1$

	3 $3$	5 $5$
	2 $2$	1 $1$

=2×(5×4-2×1)-1×(3×4-2×2)+1×(3×1-5×2) $=2 xx (5 × 4 - 2 × 1) -1 xx (3 × 4 - 2 × 2) +1 xx (3 × 1 - 5 × 2)$

=2×(20-2)-1×(12-4)+1×(3-10) $=2 xx (20 -2) -1 xx (12 -4) +1 xx (3 -10)$

=2×(18)-1×(8)+1×(-7) $=2 xx (18) -1 xx (8) +1 xx (-7)$

=36-8-7 $= 36 -8 -7$

=21 $=21$

|D1| $|D_1|$

5 $5$	1 $1$	1 $1$
15 $15$	5 $5$	2 $2$
8 $8$	1 $1$	4 $4$

5 $5$

	5 $5$	2 $2$
	1 $1$	4 $4$

-1 $-1$

	15 $15$	2 $2$
	8 $8$	4 $4$

+1 $+1$

	15 $15$	5 $5$
	8 $8$	1 $1$

=5×(5×4-2×1)-1×(15×4-2×8)+1×(15×1-5×8) $=5 xx (5 × 4 - 2 × 1) -1 xx (15 × 4 - 2 × 8) +1 xx (15 × 1 - 5 × 8)$

=5×(20-2)-1×(60-16)+1×(15-40) $=5 xx (20 -2) -1 xx (60 -16) +1 xx (15 -40)$

=5×(18)-1×(44)+1×(-25) $=5 xx (18) -1 xx (44) +1 xx (-25)$

=90-44-25 $= 90 -44 -25$

=21 $=21$

|D2| $|D_2|$

2 $2$	5 $5$	1 $1$
3 $3$	15 $15$	2 $2$
2 $2$	8 $8$	4 $4$

2 $2$

	15 $15$	2 $2$
	8 $8$	4 $4$

-5 $-5$

	3 $3$	2 $2$
	2 $2$	4 $4$

+1 $+1$

	3 $3$	15 $15$
	2 $2$	8 $8$

=2×(15×4-2×8)-5×(3×4-2×2)+1×(3×8-15×2) $=2 xx (15 × 4 - 2 × 8) -5 xx (3 × 4 - 2 × 2) +1 xx (3 × 8 - 15 × 2)$

=2×(60-16)-5×(12-4)+1×(24-30) $=2 xx (60 -16) -5 xx (12 -4) +1 xx (24 -30)$

=2×(44)-5×(8)+1×(-6) $=2 xx (44) -5 xx (8) +1 xx (-6)$

=88-40-6 $= 88 -40 -6$

=42 $=42$

|D3| $|D_3|$

2 $2$	1 $1$	5 $5$
3 $3$	5 $5$	15 $15$
2 $2$	1 $1$	8 $8$

2 $2$

	5 $5$	15 $15$
	1 $1$	8 $8$

-1 $-1$

	3 $3$	15 $15$
	2 $2$	8 $8$

+5 $+5$

	3 $3$	5 $5$
	2 $2$	1 $1$

=2×(5×8-15×1)-1×(3×8-15×2)+5×(3×1-5×2) $=2 xx (5 × 8 - 15 × 1) -1 xx (3 × 8 - 15 × 2) +5 xx (3 × 1 - 5 × 2)$

=2×(40-15)-1×(24-30)+5×(3-10) $=2 xx (40 -15) -1 xx (24 -30) +5 xx (3 -10)$

=2×(25)-1×(-6)+5×(-7) $=2 xx (25) -1 xx (-6) +5 xx (-7)$

=50+6-35 $= 50 +6 -35$

=21 $=21$

Here

D=21,D1=21,D2=42,D3=21 $D=21,D_1=21,D_2=42,D_3=21$

Here,

D≠0 $D!=0$

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

2. Determine whether the system of linear equations 2x+5y=16,4x+10y=32 is inconsistent

Solution:
Here

2x+5y=16 $2x+5y=16$

4x+10y=32 $4x+10y=32$

Comparing

2x+5y=16 $2x+5y=16$ with

a1x+b1y+c1=0 $a_1x+b_1y+c_1=0$

we get

a1=2,b1=5,c1=-16 $a_1=2,b_1=5,c_1=-16$

Comparing

4x+10y=32 $4x+10y=32$ with

a2x+b2y+c2=0 $a_2x+b_2y+c_2=0$

we get

a2=4,b2=10,c2=-32 $a_2=4,b_2=10,c_2=-32$

a1a2=24=12 $a_1/a_2=(2)/(4)=1/2$

b1b2=510=12 $b_1/b_2=(5)/(10)=1/2$

c1c2=1632=12 $c_1/c_2=(16)/(32)=1/2$

a1a2=b1b2=c1c2 $a_1/a_2=b_1/b_2=c_1/c_2$

So the given system has infinite solution (System of equation is consistent)

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1. Examples