For the system of linear equations x + y + z = 6 α x + β y + 7 z = 3 x + 2 y + 3 z = 14 which of…

For the system of linear equations
x+y+z=6

αx+βy+7z=3

x+2y+3z=14
which of the following is NOT true ?

  1.  If α=β=7, then the system has no solution
  2.  If α=β and α≠7 then the system has a unique solution.
  3. There is a unique point (α,β) on the line x+2y+18=0 for which the system has infinitely many solutions
  4. For every point (α,β)≠(7,7) on the line x-2y+7=0, the system has infinitely many solutions.

Solution

Given system of linear equations:

x+y+z=6

αx+βy+7z=3

x+2y+3z=14

On taking determinant, we get

A=111αβ7123

=(3β-14)-(3α-7)+(2α-β)

=2β-α-7

Here, if α=β=7 then A=0 and the system has no solution. So, option A is correct.

If α=β, α≠7 then the system has a unique solution. So, option B is correct.

If (α, β)≠(7, 7) then A≠0. So, system has not infinitely many solution. Therefore, option D is false.

Asked in: JEE Main 2023 (31 Jan Shift 1)

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