Find the Variables

Problem

[[10],[−15],[131]]=a*[[1],[−1],[8]]+b*[[2],[1],[7]]+c*[[1],[3],[−4]]+d*[[3],[−4],[36]]

Solution

1. Write the system of linear equations by equating the components of the vectors.

a+2*b+c+3*d=10

−a+b+3*c−4*d=−15

8*a+7*b−4*c+36*d=131

2. Eliminate a from the second and third equations by adding the first equation to the second, and subtracting 8 times the first from the third.

3*b+4*c−d=−5

−9*b−12*c+12*d=51

3. Simplify the third equation by adding 3 times the new second equation to it.

3*(3*b+4*c−d)+(−9*b−12*c+12*d)=3*(−5)+51

9*d=36

4. Solve for d

d=4

5. Substitute d=4 back into the equation 3*b+4*c−d=−5 to find a relationship between b and c

3*b+4*c−4=−5

3*b+4*c=−1

b=(−1−4*c)/3

6. Substitute the expressions for d and b into the first equation to solve for a in terms of c

a+2*((−1−4*c)/3)+c+3*(4)=10

a−2/3−(8*c)/3+c+12=10

a−(5*c)/3=−2+2/3

a=(5*c−4)/3

7. Identify that since there are three equations and four variables, the system has infinitely many solutions parameterized by c

Final Answer

[[a],[b],[b],[d]]=[[(5*c−4)/3],[(−4*c−1)/3],[c],[4]]

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