Write as a Vector Equality x-4y+z=2 , y=5x-z+2 , 4x+2z=2

Problem

{[x−4*y+z=2],[y=5*x−z+2],[4*x+2*z=2])

Solution

1. Rearrange the equations into standard form a*x+b*y+c*z=d so that all variables are on the left side and constants are on the right.

x−4*y+z=2

−5*x+y+z=2

4*x+0*y+2*z=2

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

Coefficients of *x:[[1],[−5],[4]]

Coefficients of *y:[[−4],[1],[0]]

Coefficients of *z:[[1],[1],[2]]

3. Construct the vector equation by multiplying each variable by its corresponding coefficient vector and setting the sum equal to the constant vector.

x*[[1],[−5],[4]]+y*[[−4],[1],[0]]+z*[[1],[1],[2]]=[[2],[2],[2]]

Final Answer

x*[[1],[−5],[4]]+y*[[−4],[1],[0]]+z*[[1],[1],[2]]=[[2],[2],[2]]

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