Find the Variables

Problem

[[1,1,0],[−3,0,1],[0,−2,1]]*[[b],[d],[T]]=[[0],[3*g],[29]]

Solution

1. Write the system of linear equations from the matrix multiplication.

b+d=0

−3*b+T=3*g

−2*d+T=29

2. Solve the first equation for d in terms of b

d=−b

3. Substitute the expression for d into the third equation to express T in terms of b

−2*(−b)+T=29

2*b+T=29

T=29−2*b

4. Substitute the expression for T into the second equation to solve for b

−3*b+(29−2*b)=3*g

−5*b+29=3*g

−5*b=3*g−29

b=(29−3*g)/5

5. Calculate the value of d using the relationship d=−b

d=−(29−3*g)/5

d=(3*g−29)/5

6. Calculate the value of T using the expression T=29−2*b

T=29−2*((29−3*g)/5)

T=(145−58+6*g)/5

T=(87+6*g)/5

Final Answer

[[b],[d],[T]]=[[(29−3*g)/5],[(3*g−29)/5],[(87+6*g)/5]]

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