Find the Variables

Problem

[[3*a−b+c,4*b+d],[−3*a+5*c−d,−3*b+6*d]]=[[5,6],[7,−3]]

Solution

1. Set up a system of linear equations by equating the corresponding elements of the two matrices.

3*a−b+c=5

4*b+d=6

−3*a+5*c−d=7

−3*b+6*d=−3

2. Solve for b and d using the second and fourth equations.

d=6−4*b

−3*b+6*(6−4*b)=−3

−3*b+36−24*b=−3

−27*b=−39

b=13/9

3. Substitute the value of b back to find d

d=6−4*(13/9)

d=54/9−52/9

d=2/9

4. Add the first and third equations to eliminate a

(3*a−b+c)+(−3*a+5*c−d)=5+7

−b+6*c−d=12

5. Substitute the known values of b and d into the combined equation to find c

−13/9+6*c−2/9=12

6*c−15/9=12

6*c=12+5/3

6*c=41/3

c=41/18

6. Substitute the values of b and c into the first equation to find a

3*a−13/9+41/18=5

3*a−26/18+41/18=5

3*a+15/18=5

3*a=5−5/6

3*a=25/6

a=25/18

Final Answer

a=25/18,b=13/9,c=41/18,d=2/9

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