InterviewSolution
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InterviewSolution
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consider the system of equations : ltbr. `3x-y +4z=3` `x+2y-3z =-2` `6x+5y+lambdaz =-3` Prove that system of equation has at least one solution for all real values of `lambda`.also prove that infinite solutions of the system of equations satisfy `(7x-4)/(-5)=(7y+9)/(13)=z` |
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Answer» `Delta= |{:(3,,-1,,4),(1,,2,,-3),(6,,5,,lambda):}|=7lambda +35` `" If "7lambda +35ne 0 i.e., lambda ne -5` then system has a unique solution (As `Delta ne 0` gives unique solution). But if `lambda =-5` then we have `Delta =0` . Solution exists in this case if `Delta_(x) =Delta_(y) =Delta_(z)=0` `" For " lambda =-5` `Delta_(x)= |{:(3,,-1,,4),(-2,,2,,-3),(-3,,5,,5):}|=0` `Delta_(y)= |{:(3,,-3,,4),(1,,-2,,-3),(6,,-3,,-5):}|=0` `" and "Delta_(z)= |{:(3,,-1,,3),(1,,2,,-2),(6,,5,,-3):}|=0` Thus `Delta =Delta_(x)=Delta_(y)=Delta_(z)=0` and hence there exists infinite number of solutions. Now. eliminating x from the equations .we get `7y-13z =-9` Let us put `z = k in R. :. y= (13k -9)//7" and " so x=(4-5k)//7` Where k is any real number . `" Thus " .(7x-4)/(-5) =(7y+9)/(13)=z` |
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