Write as a Vector Equality x+y-z=4 , x-2y+2z=-5 , 2x-y+2z=-1

Problem

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

Solution

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

2. Extract the coefficients of x from each equation to create the first vector:

([1],[1],[2])

3. Extract the coefficients of y from each equation to create the second vector:

([1],[−2],[−1])

4. Extract the coefficients of z from each equation to create the third vector:

([−1],[2],[2])

5. Identify the constants on the right side of the equations to form the constant vector:

([4],[−5],[−1])

6. Combine these into a single vector equality where the variables scale their respective coefficient vectors.

Final Answer

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

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