Question Number 124071 by zarminaawan last updated on 30/Nov/20
$${Given}\:{the}\:{system}\:{of}\:{linear}\:{equation} \\ $$$${x}_{\mathrm{1}} +\mathrm{2}{x}_{\mathrm{2}} −\mathrm{3}{x}_{\mathrm{3}} =\mathrm{4} \\ $$$$\mathrm{3}{x}_{\mathrm{1}} −{x}_{\mathrm{2}} +\mathrm{5}{x}_{\mathrm{3}} =\mathrm{2} \\ $$$$\mathrm{4}{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +\left({a}^{\mathrm{2}} −\mathrm{14}\right){x}_{\mathrm{3}} ={a}+\mathrm{2} \\ $$$${for}\:{which}\:{value}\:{of}\:“{a}''\:{the}\:{system}\:{have}\:{one}\:{solution},{infinite}\:{sol}.{and}\:{number}\:{solution}. \\ $$$$ \\ $$
Commented by danishnaveed last updated on 30/Nov/20
$${send}\:{ans} \\ $$
Answered by mr W last updated on 30/Nov/20
$$\left(\mathrm{1}\right)+\left(\mathrm{2}\right): \\ $$$$\mathrm{4}{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +\mathrm{2}{x}_{\mathrm{3}} =\mathrm{6}\:\:\:…\left(\mathrm{4}\right) \\ $$$$\mathrm{4}{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +\left({a}^{\mathrm{2}} −\mathrm{14}\right){x}_{\mathrm{3}} ={a}+\mathrm{2}\:\:\:\:…\left(\mathrm{3}\right) \\ $$$$\left(\mathrm{4}\right)−\left(\mathrm{3}\right): \\ $$$$\left(\mathrm{4}+{a}\right)\left(\mathrm{4}−{a}\right){x}_{\mathrm{3}} =\mathrm{4}−{a}\:\:\:…\left(\mathrm{5}\right) \\ $$$${if}\:{a}=\mathrm{4}\:\Rightarrow\:{infinite}\:{solutions} \\ $$$${if}\:{a}=−\mathrm{4}\:\Rightarrow\:{no}\:{solution} \\ $$$${if}\:{a}\neq\mathrm{4}\:\&\:{a}\neq−\mathrm{4}\:\Rightarrow\:{one}\:{solution}\:{x}_{\mathrm{3}} =\frac{\mathrm{1}}{{a}+\mathrm{4}} \\ $$