The values of λ for which the system of equations $$x+y$$−$$3=0$$,$$1+$$λ$$x+2+$$λy−$$8$$,x−$$1+$$λ$$y+2+$$λ$$=0$$ is consistent are

Question

The values of λ  for which the system of equations $$x+y$$−$$3=0$$,$$1+$$λ$$x+2+$$λy−$$8$$,x−$$1+$$λ$$y+2+$$λ$$=0$$  is consistent are

Infinity Learn · Accepted Answer

The given system of equations will be consistent if Δ$$=11$$−$$31+$$λ$$2+$$λ−$$81$$−$$(1+$$λ$$)2+$$λ$$=0$$ Applying $$C2$$→$$C2$$−$$C1$$ and $$C3$$→$$C3+3C1$$, we get Δ$$=1001+$$λ$$1$$−$$5+3$$λ$$1$$−$$2$$−λ$$5+$$λ$$=0$$⇒$$(5+$$λ$$)+(2+$$λ$$)(3$$λ−$$5)=0$$⇒$$5+$$λ$$+6$$λ−$$10+3$$λ$$2$$−$$5$$λ$$=0$$⇒$$3$$λ$$2+2$$λ−$$5=0$$⇒$$(3$$λ$$+5)($$λ−$$1)=0$$⇒λ$$=$$−$$5/3$$ or λ$$=1$$

The values of $λ$ for which the system of equations $x + y - 3 = 0, (1 + λ) x + (2 + λ) y - 8, x - (1 + λ) y + (2 + λ) = 0$ is consistent are

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The values of λ for which the system of equations x+y−3=0,1+λx+2+λy−8,x−1+λy+2+λ=0 is consistent are

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The values of $λ$ for which the system of equations $x + y - 3 = 0, (1 + λ) x + (2 + λ) y - 8, x - (1 + λ) y + (2 + λ) = 0$ is consistent are