Find the Variables

Problem

[[a−b,b+c],[3*d+c,2*a−4*d]]=[[13,2],[12,10]]

Solution

1. Set up a system of equations by equating corresponding entries of the matrices.

a−b=13

b+c=2

3*d+c=12

2*a−4*d=10

2. Express b and c in terms of a and d using the first and third equations.

b=a−13

c=12−3*d

3. Substitute these expressions into the second equation to create a relationship between a and d

(a−13)+(12−3*d)=2

a−3*d−1=2

a−3*d=3

a=3*d+3

4. Substitute the expression for a into the fourth equation to solve for d

2*(3*d+3)−4*d=10

6*d+6−4*d=10

2*d=4

d=2

5. Solve for the remaining variables using back-substitution.

a=3*(2)+3=9

b=9−13=−4

c=12−3*(2)=6

Final Answer

a=9,b=−4,c=6,d=2

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