The functions `f_1(x)=3x-1`, `f_2(x)=4x` and `f_3(x)=2x` are linearly dependent. Show this by finding values of `c_1`, `c_2` and `c_3`, not all zeros, such that `c_1f_1(x)+c_2f_2(x)+c_3f_3(x)=0`.


`f_1(x)=3x-1`, `f_2(x)=4x`, `f_3(x)=2x`.

The task is to find such constants `c_1`, `c_2`, `c_3` not equal zero simultaneously such that




From this following system is obtained:


From second equation `c_1=0` and first equation gives `4c_2+2c_3=0` or `c_3=-2c_2`.

So, if `c_1=0`, `c_2=t`, then `c_3=-2t` `(t in R)` and `c_1f_1(x)+c_2f_2(x)+c_3f_3(x)=0`.

In particular, we can take `c_1=0`, `c_2=1`, `c_3=-2`.