Write as a Vector Equality x+y+z=12 , 2x-3y+4z=4 , x+z=2y

Problem

{[x+y+z=12],[2*x−3*y+4*z=4],[x+z=2*y])

Solution

1. Rearrange the third equation so that all variables are on the left side and the constant is on the right.

x−2*y+z=0

2. Identify the coefficients of the variables x y and z for each equation to form the column vectors.

Equation 1: *1,1,1

Equation 2: *2,−3,4

Equation 3: *1,−2,1

3. Express the system as a linear combination of vectors by multiplying each variable by its corresponding column of coefficients.

x*[[1],[2],[1]]+y*[[1],[−3],[−2]]+z*[[1],[4],[1]]=[[12],[4],[0]]

Final Answer

x*[[1],[2],[1]]+y*[[1],[−3],[−2]]+z*[[1],[4],[1]]=[[12],[4],[0]]

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