can you express 14 17 ? 23 as a linear combination of v?
v= (2 1 -1), (1 -1 2), (4 5 -7)

No ,we cannot express 14,17,23 as a linear combination of V

Step-by-step explanation:

Here 14,17,23 can be written as linear combination of V if we have a,b and c all non zero such that

(14,17,23) = a(2,1,-1)+b(1,-1,2)+c(4,5,-7)

(14,17,23) = (2a,a,-a)+(b,-b,2b)+(4c,5c,-7c)   [ scalar multiplication ]

(14,17,23)= (2a+b+4c,a-b+5c,-a+2b-7c)    [ vector addition ]

2a+b+4c = 14               equation 1

a-b+5c= 17                  equation 2

-a+2b-7c =23             equation 3

Here we  have three simultaneous equations

Since equation 2 and 3 has sign of a is opposite  with coefficient 1,adding both equations,we get

b-2c = 40

multiplying equation equation 3 by 2 and adding it to equation 1 in order to eliminate the term a

2a+b+4c=14

-2a+4b-14c=46

adding 5b-10c = 60

           divide it both sides by 5 ,we get

              b -2c = 12

here we see that  left side of two equations same but right side is not same

therefore solution doesnt exist .

Moreover given set of Vectors in V are not linear independent

since their determinant is zero

in order to write linear combination ,we need to have linearly independent vectors

there exist not values of a  ,and c

therefore we cannot express 14,17,23 as linear combination of V

System:
 

x+y=11

x=11-y then 11-y+1.75y=14
0.75y=14-11=3
y=4 - orange juice
x=11-4=7 - apple juice

if you wanna use a graphic method you should draw the lines of each equation. The intersection of these lines is solution