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

Problem

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

Solution

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

2. Extract the coefficients of x from each equation to create the first vector: ([1],[2],[1])

3. Extract the coefficients of y from each equation to create the second vector: ([2],[1],[2])

4. Extract the coefficients of z from each equation to create the third vector: ([−1],[1],[1])

5. Identify the constants on the right side of the equations to form the result vector: ([4],[−2],[2])

6. Combine these into a single vector equation where each variable scales its corresponding coefficient vector.

Final Answer

x*([1],[2],[1])+y*([2],[1],[2])+z*([−1],[1],[1])=([4],[−2],[2])

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