show that the system of equation is consistent & hence solve it. x+2y+z=2 3x+y-2z=1 4x-3y-z=3 2x-4y+2z=4
Answers
Step-by-step explanation:
Let's check the value of △,△
1
,△
2
,△
3
for knowing whether system of linear equations is consistent or not,
△=
∣
∣
∣
∣
∣
∣
∣
∣
2
4
6
2
4
6
−2
−1
2
∣
∣
∣
∣
∣
∣
∣
∣
=0 as Determinant is 0 if two of its rows or columns are same .
△
1
=
∣
∣
∣
∣
∣
∣
∣
∣
1
2
3
2
4
6
−2
−1
2
∣
∣
∣
∣
∣
∣
∣
∣
=2
∣
∣
∣
∣
∣
∣
∣
∣
1
2
3
1
2
3
−2
−1
2
∣
∣
∣
∣
∣
∣
∣
∣
=0 as Determinant is 0 if two of its rows or columns are same .
△
2
=
∣
∣
∣
∣
∣
∣
∣
∣
2
4
6
1
2
3
−2
−1
2
∣
∣
∣
∣
∣
∣
∣
∣
=2
∣
∣
∣
∣
∣
∣
∣
∣
1
2
3
1
2
3
−2
−1
2
∣
∣
∣
∣
∣
∣
∣
∣
=0 as Determinant is 0 if two of its rows or columns are same .
△
3
=
∣
∣
∣
∣
∣
∣
∣
∣
2
4
6
2
4
6
1
2
3
∣
∣
∣
∣
∣
∣
∣
∣
=0 as Determinant is 0 if two of its rows or columns are same .
Since, △=△
1
=△
2
=△
3
=0, the given system of equations have infinite number of solutions and thus is consistent .
Let x =k,
Then, 6y+2z=3−6k
4y−z=2−4k
Simplifying we get,
y=
2
1
−k & z=0