Write as a Vector Equality x-z+3y=4 , z=3y , y-x=5z

Problem

x−z+3*y=4,z=3*y,y−x=5*z

Solution

1. Rearrange each equation into the standard form a*x+b*y+c*z=d so that the variables x y and z are aligned.

x+3*y−z=4

0*x−3*y+z=0

−x+y−5*z=0

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

Coefficients of *x:([1],[0],[−1])

Coefficients of *y:([3],[−3],[1])

Coefficients of *z:([−1],[1],[−5])

3. Construct the vector equality by multiplying each variable by its corresponding coefficient vector and setting the sum equal to the constant vector from the right-hand side.

x*([1],[0],[−1])+y*([3],[−3],[1])+z*([−1],[1],[−5])=([4],[0],[0])

Final Answer

x*([1],[0],[−1])+y*([3],[−3],[1])+z*([−1],[1],[−5])=([4],[0],[0])

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