Find the Variables

Problem

[[1,0,0],[1,−1,1],[1,2,4],[1,1,1]]*[[x],[y],[z]]=[[a],[b],[c],[d]]

Solution

1. Perform matrix multiplication on the left side to create a system of linear equations.

x=a

x−y+z=b

x+2*y+4*z=c

x+y+z=d

2. Substitute the first equation x=a into the remaining three equations.

a−y+z=b

a+2*y+4*z=c

a+y+z=d

3. Isolate the variables in the simplified system.

−y+z=b−a

2*y+4*z=c−a

y+z=d−a

4. Solve for y and z using the first and third equations of the simplified system by adding them together.

(−y+z)+(y+z)=(b−a)+(d−a)

2*z=b+d−2*a

z=(b+d−2*a)/2

5. Solve for y by subtracting the first simplified equation from the third.

(y+z)−(−y+z)=(d−a)−(b−a)

2*y=d−b

y=(d−b)/2

6. Check for consistency using the second simplified equation 2*y+4*z=c−a

2*((d−b)/2)+4*((b+d−2*a)/2)=c−a

(d−b)+(2*b+2*d−4*a)=c−a

3*d+b−4*a=c−a

3*d+b−3*a−c=0

Final Answer

[[x],[y],[z]]=[[a],[(d−b)/2],[(b+d−2*a)/2]]* provided *3*a−b+c−3*d=0

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