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`

Explanation

`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`