Find the Variables

Problem

a*[[1],[2],[−1]]+b*[[1],[3],[−2]]+c*[[3],[7],[−4]]=[[0],[0],[0]]

Solution

1. Set up the system of linear equations by multiplying the scalars into the vectors and adding the components.

a+b+3*c=0

2*a+3*b+7*c=0

−a−2*b−4*c=0

2. Create an augmented matrix to represent the system.

[[1,1,3,0],[2,3,7,0],[−1,−2,−4,0]]

3. Perform row operations to reach row-echelon form. Subtract 2 times the first row from the second row, and add the first row to the third row.

[[1,1,3,0],[0,1,1,0],[0,−1,−1,0]]

4. Simplify the matrix further by adding the second row to the third row.

[[1,1,3,0],[0,1,1,0],[0,0,0,0]]

5. Identify the free variable. Since the third row is all zeros, c is a free variable. Let c=t

c=t

6. Solve for b using the second row.

b+t=0

b=−t

7. Solve for a using the first row.

a+(−t)+3*t=0

a+2*t=0

a=−2*t

Final Answer

a=−2*t,b=−t,c=t

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